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Principle Of Mathematical Induction - Class 11 Engineering Maths - Extra Questions

If A=(cosθisinθisinθcosθ), where i=1, then by the principle of Mathematics Induction prove that An=(cosnθisinnθisinnθcosnθ)



Using the principle  mathematical induction , prove that nϵN
In=π/20cosnxsinnxdx
=12n+1[2+222+233....+2nn] 



Prove that 2n>n for all positive integers n



For any natural number n  , prove that inequality 
|sin n x| n | sin x | 



Prove that 7 is a factor of 23n - 1 for all natural numbers n.



Prove the assertions of the following problems 
Prove that 2^n >  n^2 for any natural number n > 5. 



If P(n) is the statement ''2^n\ge 3n" and if P(r) is true prove that P(r+1) is true.



Using the principle of mathematical induction prove that 2 + 4 + 6 + .... + 2n = {n^2} + n.



Prove that \left( {1 \times 2 \times 5} \right) + \left( {2 \times 3 \times 7} \right) + \left( {3 \times 4 \times 9} \right) + ..... is \dfrac{{n\left( {n + 1} \right)\left( {n + 2} \right)\left( {3n + 7} \right)}}{6}



Prove by mathematical induction,
{ 1 }^{ 2 }+{ 2 }^{ 2 }+{ 3 }^{ 2 }+....+{ n }^{ 2 }=\dfrac { n\left( n+1 \right) \left( 2n+1 \right)  }{ 6 } 



Using the principle of mathematical induction, show that;
2+4+6+.....+2n=n^2+n



Using Principle of mathematical induction prove that 6^n -1 divisible by 5.



Using the principle of Mathematical Induction, prove the following for all n and N
1) {3}^{n}>{2}^{n}



Prove that the square of any positive integer of the form 5m+1 will leave a remainder 1 when divided by 5 for some integer m



Prove using PMI
\left( {1 + \frac{3}{1}} \right)\left( {1 + \frac{5}{4}} \right)\left( {1 + \frac{7}{9}} \right)...\left( {1 + \frac{{\left( {2n + 1} \right)}}{{{n^2}}}} \right) = {\left( {n + 1} \right)^2}



If P(n) is the statement n^{2}-n+41 is prime, prove that P(1),P(2) and P(3) are true.
Prove also that P(41) is not true.



Solve: (2n+7)<(n+3)^2



Using P.M.I. prove that
{n^2} < {2^n}\forall n \ge 5



x = \sin t\,\,\,\,\,\,\,\,\,\,y = \cos mt
Prove \left( {1 - {x^2}} \right){y_{n + 2}} - \left( {2n - 1} \right){y_{n + 1}} - \left( {{n^2} - {m^2}} \right){y_n} = 0



Use the principle of mathematical induction to show that :
2 + 2^2 + ..... + 2^n = 2^{n+1} - 2 for every natural number n.



Show that only one of the no n, n + 2 and n + 4 in divisible 3.



Prove by method of induction, for all n \in
1^2 + 2^2 + 3^2 + ... + n^2 = \dfrac{n(n + 1)(2n + 1)}{6}.



If {a_1},\,{a_2},\,{a_3},\,...,{a_n} are in A.P. and {a_i} > 0 for all i, then show that \dfrac{1}{{{a_1}{a_2}}} + \dfrac{1}{{{a_2}{a_3}}} + ... + \dfrac{1}{{{a_{n - 1}}{a_n}}} = \dfrac{{n - 1}}{{{a_1}{a_n}}}.



To prove: n \le 2^{n} for n\ in\  N
P(n):n \le 2^{n}......... for n\ in\  N



Can the number 6^{n}, where n is a natural number, end with digit 5? Give reasons.



For every natural number n(>1), prove that 4^{n}+15n-1 is divisible by 9



Prove by method of induction; for all n \in N  3 + 7 + 11 + \ldots . . . + to n terms = n ( 2 n + 1 )



Prove that 3^{n+1} > 3(n+1).



"The product of three consecutive positive integers is divisible by 6". Is.this statement true or false"? Justify your answer.



Prove that 1 + 2 + 3 + ..... + n = \displaystyle \frac {n(n\, +\, 1)}{2}.



Prove that 3^{n + 1} > 3(n + 1).



Prove  1+3+5+\cdots+(2n-1)=n^2



Prove 3^{n+1}>3(n+1)



Prove the following using the principle of mathematical induction for all n\in N:
1+3+{3}^{2}+........+{3}^{n-1}=\cfrac{({3}^{n}-1)}{2}



Prove that 1 + 3 + 5 + ..... + (2n - 1) = n ^2.



Prove the following by using principle of mathematical induction for all n\in  N: 1.3+3.5+5.7+.......+(2n-1)(2n+1)=\cfrac{n(4{n}^{2}+6n-1)}{3}



Prove the following by using the principle of mathematical induction for all n\in N: 1.3+2.{3}^{2}+3.{3}^{3}+.....+n.{3}{^n}=\cfrac{(2n-1){3}^{n+1}+3}{4}



Prove the following by using the principle of mathematical induction for all n\in N:1.2.3+2.3.4+......+n(n+1)(n+2)=\cfrac{n(n+1)(n+2)(n+3)}{4}



Prove the following by using the principle of mathematical induction for all n \in N: \cfrac{1}{2}+\cfrac{1}{4}+\cfrac{1}{8}+.....+\cfrac{1}{{2}^{n}}=1-\cfrac{1}{{2}^{n}}



Prove the following by using the principle of mathematical induction for all n\in N:\cfrac { 1 }{ 1.2.3 } +\cfrac { 1 }{ 2.3.4 } +\cfrac { 1 }{ 3.4.5 } +.....+\cfrac { 1 }{ n(n+1)(n+2) } =\cfrac { n(n+3) }{ 4(n+1)(n+2) }



Prove the following by using the principle of mathematical induction for all n\in :
1+\cfrac{1}{(1+2)}+\cfrac{1}{(1+2+3)}+..........+\cfrac{1}{1+2+3+.....n)}=\cfrac{2n}{(n+1)}



Prove the following by using the principle of mathematical induction for all n\in  N: 1.2+2.3+3.4+......+n(n+1)=\left[ \cfrac { n(n+1)(n+2) }{ 3 }  \right] 



Prove the following by using the principle of mathematical induction for all n\in N
{ 1 }^{ 3 }+{ 2 }^{ 3 }+{ 3 }^{ 3 }+.......+{ n }^{ 3 }={ \left[ \cfrac { n(n+1) }{ 2 }  \right]  }^{ 2 }



Prove the following by using the principle of mathematical induction for all n\in N: 1.2+2.{2}^{2}+3.{2}^{2}+.......+n.{2}^{n}=(n-1){2}^{n+1}+2



Prove the following by using the principle of mathematical induction for all n\in N: \cfrac{1}{2.5}+\cfrac{1}{5.8}+\cfrac{1}{8.11}+......+\cfrac{1}{(3n-1)(3n+2)}=\cfrac{n}{(6n+4)}



Prove the following by using the principle of mathematical induction for all n\in N: \left( 1+\cfrac { 3 }{ 1 }  \right) \left( 1+\cfrac { 5 }{ 4 }  \right) \left( 1+\cfrac { 7 }{ 9 }  \right) ......\left( 1+\cfrac { (2n+1) }{ { n }^{ 2 } }  \right) ={ (n+1) }^{ 2 }



Prove the following by using the principle of mathematical induction for all n\in N:{10}^{2n-1}+1 is divisible by 11



Prove the following by using the principle of mathematical induction for all n\in N:{x}^{2n}-{y}^{2n} is divisible by x+y.



Prove the following b y using the principle of mathematical induction for all n\in N:1+2+3+.....+n<\cfrac{1}{8}{(2n+1)}^{2}



Prove the following by using principle of mathematical induction for all n\in N: \cfrac { 1 }{ 3.5 } +\cfrac { 1 }{ 5.7 } +\cfrac { 1 }{ 7.9 } +.....+\cfrac { 1 }{ (2n+1)(2n+3) } =\cfrac { n }{ 3(2n+3) }



Prove the following by using the principle of mathematical induction for all n\in N:\cfrac { 1 }{ 1.4 } +\cfrac { 1 }{ 4.7 } +\cfrac { 1 }{ 7.10 } +.....+\cfrac { 1 }{ (3n-2)(3n+1) } =\cfrac { n }{ (3n+1) }



Prove the following by using the principle of mathematical induction for all n\in N: P(n):a+ar+a{r}^{2}+......+a{r}^{n-1}=\cfrac{a({r}^{n}-1)}{r-1}



Prove the following by using the principle of mathematical induction for all n\in N:{ 1 }^{ 2 }+{ 3 }^{ 2 }+{ 5 }^{ 2 }+.......+({ 2n-1) }^{ 2 }=\cfrac { n(2n-1)(2n+1) }{ 3 }



Prove the following by using the principle of mathematical induction for all n\in N:n(n+1)(n+5) is multiple of 3.



Prove the following by using the principle of mathematical induction for all n\in N:\left( 1+\cfrac { 1 }{ 1 }  \right) \left( 1+\cfrac { 1 }{ 2 }  \right) \left( 1+\cfrac { 1 }{ 3 }  \right) ......\left( 1+\cfrac { 1 }{ n }  \right) =(n+1)\quad



Find counter examples to disprove the following statement.
A quadrilateral with all sides are equal is a square.



Using mathematical induction, prove that \cfrac{d}{dx}({x}^{n})=n{x}^{n-1} for all positive integers n.



Prove the following by using the principle of mathematical induction for all n\in N:{41}^{n}-{14}^{n} is a multiple of 27.



Prove the following by using the principle of mathematical induction for all n\in N:(2n+7)< {(n+3)}^{2}.



Prove the following by using the principle of mathematical induction for all n\in N:{3}^{2n+2}-8n-9 is divisible by 8.



Prove by induction:
{1}^{2}+{2}^{2}+{3}^{2}+........+{n}^{2}=\cfrac{1}{6}n(n+1)(2n+1)



Prove by induction:
{x}^{n}-{y}^{n} is divisible by x+y when n is even.



Show that { 2 }^{ 2n }-3n-1 is divisible by 9 for all positive integral values of n.



Prove by induction:
2+{2}^{2}+{2}^{3}+........+{2}^{n}=2({2}^{n}-1)



Prove the following by the principle of mathematical induction: (n\epsilon N)
1^{2} . 2 + 2^{2} . 3 + .... + n^{2} (n + 1) = \dfrac {n(n + 1)(n + 2)(3n + 1)}{12}.



Prove the following by the principle of Mathematical Induction.
(2n + 1)(2n - 1) is an odd number for all n\epsilon N.



Prove by induction:
\cfrac{1}{1.2}+\cfrac{1}{2.3}+\cfrac{1}{3.4}+..... to n terms=\cfrac{n}{n+1}



Prove by induction: 
1+3+5+.....+(2n-1)={n}^{2}



If p and q are any two positive integers, show that |\underline {pq} is divisible by (|\underline {p})^{q} . |\underline {q} and by (|\underline {q})^{p} . |\underline {p.}



(1 + x)^n >1 + nx  for n > 2 , n \epsilon
x > -1  x \neq 



Prove by Mathematical induction that
{ 1 }^{ 2 }+{ 3 }^{ 2 }+{ 5 }^{ 2 }...{ \left( 2n-1 \right)  }^{ 2 }=\cfrac { n\left( 2n-1 \right) \left( 2n+1 \right)  }{ 3 } \forall n\in N



Prove  2 + 5 + 8 + 11 + ... + (3n - 1) = \dfrac{1}{2} n (3n + 1) n \epsilon N



Prove that 11^{n +2} + 12^{2n \, + \, 1} is divisible by 133 for any non-negative integral n.



Prove \dfrac {1}{1  \cdot 2\cdot  3 } \, + \, \dfrac {1}{2  \cdot 3\cdot 4 } \, + \, \dfrac {1}{n (n + 1)(n + 2)} \, = \, \dfrac{1(n + 3)}{4(n + 1)(n + 2)}



Prove the mathmatical induction that  \dfrac{1}{\log_x2 , \log_x4 }\, + \, \dfrac{1}{\log_x4 , \log_x8 } \, + \, ..... + \,\dfrac{1}{\log_x2^{n - 1} , \log_x4^n }
= \left (1- \dfrac{1}{n}  \right ) \dfrac{1}{(\log_x2)^2} , x > 0 , \neq -1  n \epsilon N



\sqrt 2  + \sqrt 2 +  \sqrt 2   + ...+ n\ terms = 2\cos \dfrac{\pi}{2^{n +1}} , n \epsilon N



Prove the assertions of the following problems 
Prove that the expression n^3 \, - \, n is divisible by 24 for any odd n



Let u_1 = 1, u_2 = 1 and u_{n \, + \, 2} \, + \, u_n \, for \, n\geq1.
Use mathematical induction to  show that 
u_n\, = \, \dfrac{1}{\sqrt{(5)}}\left [ \left ( \dfrac{1 \, + \, \sqrt{5}}{2} \right )^n \, - \, \left ( \dfrac{1 \, - \, \sqrt{5}}{2} \right )^n \right ] for all n \, \geq \, 1



For any natural number n > 1 
\dfrac {1}{n \, + \, 1} \, + \,  \dfrac {1}{n \, + \, 2}  \, + \, ..... \, + \, \dfrac {1}{2n} >  \dfrac {13}{14}



Given a_1\, = \, \dfrac{1}{2}\left ( a_0 \, + \, \dfrac{A}{a_0} \right ) \, , \,a_2\, = \, \dfrac{1}{2}\left ( a_1 \, + \, \dfrac{A}{a_1} \right ) 
and a_{n \, + \, 1}\, = \, \dfrac{1}{2}\left ( a_n \, + \, \dfrac{A}{a_n} \right ) 
Find n\geq \, 2 where a > 0, A > 0, prove that 
\dfrac{a_n \, - \, \sqrt{A}}{a_n \, + \, \sqrt{(A)}}\, = \, \left (\dfrac{a_1 \, - \, \sqrt{A}}{a_1 \, + \, \sqrt{(A)}}  \right )^{2^{n \, - \, 1}}
using mathematical induction.



 Applying the principal of mathematical induction of prove that 
\dfrac{2 \, sin \, \theta }{cos \, \theta \, + \, 3\, cos \, \theta} \, + \, \dfrac{2 \, sin \, \theta }{cos \, \theta \, + \, 5\, cos \, \theta} \, + \, \dfrac{2 \, sin \, \theta }{cos \, \theta \, + \,  cos \,(2n  \, + \, 1) \theta}
= tan ( n + 1) \theta  \, - \, tan \theta



Prove that induction that
sin x + sin 3x  + .. + sin(2n - 1)  x = \dfrac {sin^2 nx}{sin x}



Prove that mathematical induction that 
\dfrac{1}{1 \, + \, x} \, + \, \dfrac{2}{1 \, + \, x^2} \, + \, \dfrac{4}{1 \, + \, x^4} \, + \, ... \, + \, \dfrac{2^n}{1 \, + \, x^{2^n}}
= \, \dfrac{1}{1 \, - \, }\dfrac{2^{n \, + \, i}}{1 \, + \, x^{2^{n \, + \, i}}}



Observing that 1^3 \, = \, 1, \, 2^3 \, = \, 3 \, + \, 5,
3^3 = 7 + 9 + 11, 4^3 = 13 + 15 + 17 + 19
Find a general formula for the cube of natural number n and prove it by the principle of mathematical induction 



Explain the method of mathematical induction and use it to show that 11^{n \, + \, 2} \, + \, 12^{2n \, + \, 1} where n is natural number is divisible by 133



If x  \, + \, y \, = a \, + \, b , x^2 \, + \, y^2 \, = \, a^2 \, + \, b^2 then prove that the mathematical induction  that  x^n \, + \, y^n \, = \, a^n \, + \, b^n for all the natural number n . 



If y \, = \, \dfrac{log \, \, x}{x} , then prove by mathematical induction 
y_n \, = \, \dfrac{(-1)^n \, (n!)}{x^{n \, + \,1}}\left [ log \, \, x \, - \, 1 \, -\dfrac{1}{2}\, - \, ....-\, \dfrac{1}{n} \right ]



If x^3 = x + 1, then show that
x^{3n} \, = \, a_n x \, + \, b_n \, - \, c_n x^{-1}
Where a_{n \, + \, 1} \, = \, a_n b_n :b_{n \, + \, 1} \, = \, a_n \, + \, b_n \, + \, c_n
and c_{n \, + \, 1} \, = \, a_n \, + \, c_n



If x_1x_2x_3 \, .... \, x_n \ = \, 1(x_1 \, > \, 0, i \, = \, 1,2,.....n), prove that x_1 \, + \, x_2 \, + \, .....x_n \, \geq \, n \, (n \, \geq \, 2)



Prove that the mathematical induction that 
1 \, + \, \dfrac{1}{2} \, + \,  \dfrac{1}{3} \, + \, \dfrac{1}{4}  + ...\ , \dfrac {1}{2^n} \geq 1 \, + \, \dfrac {n}{2}
for each  no negative interger n 



Prove ^{2n} C _n  \, > \, \dfrac{4^n }{n + 1}



Prove \dfrac {2n!}{(n!)^2} \, > \, \dfrac {4n}{2n \, + \, 1}



Prove by induction \forall \, n \, \epsilon \, N \, \, if y \, = \, \dfrac{x}{x^2 \, + \, a^2}
then  y_n \, = \, \dfrac{(-1)^n \, n!}{a^{n \, + \,1}}sin^{n \, + \, 1} \, \, \theta \, \, cos(n \, + \, 1) \, \, \theta
where \theta \, = \, cot^{-1} \dfrac{x}{a}



If 0 < \alpha  < \, \dfrac{\pi}{4(n \, - \, 1)} where n > 1  , then prove that  \tan n \alpha  \, > \, n  \, \tan \, \alpha



Show by the mathematical induction that 
\dfrac{1}{sin 2 x } \, + \,  \dfrac{1}{sin 4 x } \, + \,  \dfrac{1}{sin 2^n x } \, = \, cot x  \, - \, cot 2^n \, x



Prove that { 1 }^{ 2 }+{ 2 }^{ 2 }+...+{ n }^{ 2 }>\cfrac { { n }^{ 3 } }{ 3 } ,n\epsilon N



Prove the following by using the first principle 
1+3 + 3^2 + ......+ 3^{n-1} =\frac{(3^n -1)}{2}.



Prove the following by PMI for all n belong to N
1\times3+3\times5+5\times7 .........+(2n-1)(2n+1)=\dfrac{n(4n^2+6n-1)}{3}



Prove that for any positive interger n, n^3 -n is divisible by 6.



Let p\geq 3 be an integer and \alpha , \beta be the roots of x^2 - (p + 1)x + 1 = 0.Using mathematical induction show that \alpha^n \, + \, \beta^n 
is not divisible by p.



Prove by induction method that for all n\geq  1
\int x^ne^x\, dx \, = \, n\, !\, e^x\left [ \dfrac{x^n}{n!} \, - \, \dfrac{x^{n \, - \, 1}}{(n \, - \, 1)!} \, + \, \dfrac{n^{n \, - \, 2!}}{(n \, - \, 2)!} \, + \, .....\, + \, (-1)^n \right ]



Let p\geq 3 be an integer and \alpha , \beta be the roots of x^2 - (p + 1)x + 1 = 0.Using mathematical induction show that \alpha^n \, + \, \beta^n 
is an integer 



Let  I_m \, = \, \int_{0}^{\pi}\dfrac{1 \, - \, cos \, \, mx}{1 \, - \, cos \, \, x}dx
Use mathematical induction to prove that 
I_m \, = \, m\pi \, , \, m0 , , 1, 2.....



By PMC, prove that inequality n<2^n for all n\in N



Use mathematical induction to prove
1 + 2 + 3 +  \ldots  + n < \frac{1}{8}{\left( {2n + 1} \right)^2}



If c>a; show that cba-abc=99(c-a).



if a>c; show that abc-cba=99(a-c)



Using the Mathematical induction, show that for any number n \geq 2,
\dfrac{1}{1 + 2} + \dfrac{1}{1 + 2 + 3} + \dfrac{1}{1 + 2 + 3 + 4} + ..... +\dfrac{1}{1 + 2 + 3 + ... + n} = \dfrac{n - 1}{n + 1}



Let by mathematical induction,  for any natural numbers n,
\dfrac{1}{2.5} + \dfrac{1}{5.8} + \dfrac{1}{8.11} + ..... + \dfrac{1}{(3n - 1)(3n + 2)} = \dfrac{n}{an + b}. Find a+b



Prove that {2.7^n} + {3.5^n} - 5 is divisible by 24 for all n \in N.



Prove by the principle of mathematical induction that 2^n > n for all n \in N. 



Prove the following by mathematical induction 1 + 2 + 3 +  \cdots  + n < \frac{1}{8}{\left( {2n + 1} \right)^2}



Prove that \displaystyle \frac{1}{{1 \cdot 3}} + \frac{1}{{3 \cdot 5}} + \frac{1}{{5 \cdot 7}} +  \ldots  \ldots  + \frac{1}{{\left( {2n - 1} \right)\left( {2n + 1} \right)}} = \frac{n}{{2n + 1}}



Let P(n)=n(n+1)(n+2) is divisible by 12 then which one of the following is not true.
i) P(3)
ii) P(4)
ii) P(5)
iv) P(8)



Using PMI, prove that {3^{2n + 2}} - 8n - 9 is divisible by 64.



Use mathematical induction to prove: 
\dfrac{1}{{1 \cdot 3}} + \dfrac{1}{{3 \cdot 5}} + \dfrac{1}{{5 \cdot 7}} +  \ldots  + \dfrac{1}{{\left( {2n - 1} \right)\left( {2n + 1} \right)}} = \dfrac{n}{{2n + 1}}



Prove using PMI: \displaystyle \frac{1}{{1.2.3}} + \frac{1}{{2.3.4}} + \frac{1}{{3.4.5}} + ... + \frac{1}{{n\left( {n + 1} \right)\left( {n + 2} \right)}} = \frac{{n\left( {n + 3} \right)}}{{4\left( {n + 1} \right)\left( {n + 2} \right)}}



Prove : 1 + 2 + 3 + ... + n = \dfrac{1}{2} n (n + 1)



Find m if the following equation holds true  1.2 + 2.3 + 3.4 + .. + n (n + 1) = \dfrac{1}{m} n (n + 1) (n + 2)



Prove that by mathematical induction {2^{3n}} - 1 is divisible by 7 for all natural numbers. 



Find m if the following equation holds true  2 + 4 + 6 + ... + 2n = m^2\times n (n + 1)



Prove that product of two consecutive natural number cannot be 441



Find m if the following equation holds true 1.3 + 2.4 + 3.5 + ... + n (n + 2) = \dfrac{1}{m} n (n + 1)(2n + 7)



Prove that : 1 + 3 + 3^2 + ... + 3^{n-1} = \dfrac{(3^n - 1)}{2}



Prove 1.3 + 3.5 + 5.7 + ...+ (2n -1)(2n + 1) = \dfrac{n(4n^2 + 6n -1)}{3}



Prove that 1^2 + 2^2 + 3^2 + .... + n^2 > \dfrac{n^2}{3} for all n \in N using principle of mathematical induction.



Prove by PMI:
cos \theta + cos 2\theta + ..... + cosn \theta = sin\dfrac{n\theta}{2}.cos \theta c\dfrac{\theta}{2}.cos\dfrac{(n+1)\theta}{2} 



For any odd integer n\geq 1Find the value of 
n^3-(n-1)^3+...........+(-1)^{n-1}1^3.



Prove by Induction a+ar+{ ar }^{ 2 }+..... up to n terms =\cfrac { a\left( { r }^{ n }-1 \right) }{ r-1 } ,r\neq 1



Prove that {25^n} - {20^n} - {8^n} + {3^n} is divisible by 5.



Prove that if either of 2a+3b and 9a+5b is divisible by 17 ,so is the other ,a,b \in N



Solve \dfrac{1}{{1.2.3}} + \dfrac{1}{{2.3.4}} + \dfrac{1}{{3.4.5}} +  \cdots  + \dfrac{1}{{n\left( {n + 1} \right)\left( {n + 2} \right)}} = \dfrac{{n\left( {n + 3} \right)}}{{4\left( {n + 1} \right)\left( {n + 2} \right)}}



Suppose m,n are integers and m=n^2-n. Then show that m^2-2m is divisible by 24.



If A=\begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\1 & 1 & 1 \end{bmatrix}, prove that

A^n=\begin{bmatrix} 3^{n-1} & 3^{n-1} & 3^{n-1} \\ 3^{n-1} & 3^{n-1} & 3^{n-1} \\ 3^{n-1} & 3^{n-1} & 3^{n-1} \end{bmatrix}, n\in N.



Prove that 1.2 + 2.3 +.......n(n+1) =\frac{{n(n + 1)(n + 2)}}{3} in mathematical induction.



Prove that 2.7^{n}+3.5^{n}-5 is divisible by 24, for all n\in N.



Prove by method of induction, for all n \in N 
2 + 4 + 6 + .... + 2n = n\left( {n + 1} \right)



prove the inequalities {\left( {n!} \right)^2} \le {n^n}\left( {n!} \right) < \left( {2n} \right)! for all positive integers n .



Let P(n) be the statement 3^{n} > n^{n}. If P(n) is true, prove that P(n+1) is true.



Show that 1^{2}+(1^{2}+2^{2})+(1^{2}+2^{2}+3^{2})+.... up to n terms=\dfrac{n(n+1)^{2}}{12}, n\epsilon N



{7^{2n}} + {2^{3n - 3}}{.3^{n - 1}} is divisible by 25 for any natural number n \ge 1. Prove that by mathematical 



What must be added to {x^4} + 2{x^3} - 2{x^2} + x - 1 . So that result is exactly divisible by {x^2} + x - 2 ?



From principle of mathematical induction prove that
1+2+3+....+n=\dfrac{1}{2}n(n+1)



If 33! is divisible by 2^n, then find the maximum value of n.



If A=\begin{bmatrix} \cos { \theta  }  & \sin { \theta  }  \\ -\sin { \theta  }  & \cos { \theta  }  \end{bmatrix} then prove that { A }^{ n }=\begin{bmatrix} \cos { n\theta  }  & \sin { n\theta  }  \\ -\sin { n\theta  }  & \cos { n\theta  }  \end{bmatrix}, n\in N.



(n+1)(n+2) is an even number



Show that any positive odd integers is of the form 6q+1,6q+3 or 6q+5, where q is some integer.



Use the principle of mathermatical induction to prove that
{1}^{3}+{2}^{3}+.....{n}^{3}=\cfrac{1}{4}{n}^{2}{(n+1)}^{2} for every natural number n.



Given an example of a statement P(n) which is true for all n\ge 4 but P(1), P(2) and P(3) are not true. Justify your answer.



Prove that n^{2}-n in divisible by 2 for any positive length n.



Prove that 1+2+3+.....+n=\dfrac{n(n+1)}{2}. for n being a natural numbers. 



If (27)^{999} is divided by 7, then the remainder is :



Prove by the principle of mathematical induction that for all n \in N:
1 + 4+ 7 +.... + (3n - 2) = \dfrac{1}{2}n (3n - 1)



Using mathematical induction prove that:
\dfrac { 1 }{ 1.2.3 } +\dfrac { 1 }{ 2.3.4 } +\dfrac { 1 }{ 3.4.5 } +.....+\dfrac { 1 }{ n\left( n+1 \right) \left( n+2 \right)  } =\dfrac { n\left( n+3 \right)  }{ 4\left( n+1 \right) \left( n+2 \right)  } .



Prove that n^{2}-n is divisible  by 2 for every positive integer.



1 ^ { 2 } + 2 ^ { 2 } + \ldots + n ^ { 2 } > \frac { n ^ { 3 } } { 3 } , n \in \mathbf { N }



For every +ve integer 'n' prove that 7^n-3^n is divisible by 4.



Use the principle of mathematical induction to prove that a+(a+d)+(a+2d)+ .... +[a+(n-1)d]=n/2[2a+(n-1)d]



Show that n(n+1)(2n+1) is multiple of 6 for every natural number n.



By the principle of Mathematical induction, prove that, for  n \geq 1.
            1 ^ { 2 } + 2 ^ { 2 } + 3 ^ { 2 } + \cdots + n ^ { 2 } > \dfrac { n ^ { 3 } } { 3 }.



Prove for n\in \mathbb{N}.
1.3 + {2.3^2} + {3.3^3} + ... + n{.3^n} = \dfrac{{\left( {2n - 1} \right){3^{n + 1}} + 3}}{4}



Prove the following by using principle of mathematical induction for all  n \in N
1+\dfrac { 1 }{ (1+2) } +\dfrac { 1 }{ (1+2+3) } +.......+\dfrac { 1 }{ (1+2+3+\dots n) } =\dfrac { 2n }{ (n+1) } .



Prove the following by using the principle of mathematical induction for all n\in N: 
1+3+{ 3 }^{ 2 }+...+{ 3 }^{ n-1 }=\dfrac { \left( { 3 }^{ n }-1 \right)  }{ 2 }



Solve:9\le 1-2x



Prove: \left( {2n + 7} \right) < {\left( {n + 3} \right)^2}. for n\in\mathbb{N}.



Using Principle of Mathematical Induction, prove that
 1.3 + 2.3 ^ { 2 } + 3.3 ^ { 3 } + \ldots . n .3 ^ { n } = \dfrac { ( 2 n - 1 ) 3 ^ { n + 1 } + 3 } { 4 } ,  where  n \in \mathrm { N }



In triangle ABC, D and E are the points on the sides AB and AC. DE||BC, BD=5 cm and DE:BC=2:Find AD and the ratio of are of triangle ADE to the quadrilateral BCED.



Using principle of Mathematical induction prove that:
x ^ { 2 n } - y ^ { 2 n } is divisible by x + y, where n \in N



Using the principle of mathematical induction, prove the following for all n\in N:
2+6+18+...+2.3^{n-1}=(3^n-1).



Using the principle of mathematical induction, prove the following for all n\in N:
\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{8}+...+\dfrac{1}{2^n}=\left(1-\dfrac{1}{2^n}\right).



Using the principle of mathematical induction, prove the following for all n\in N:
1+2+3+4+....+n=\dfrac{1}{2}n(n+1).



Using the principle of mathematical induction, prove the following for all n\in N:
1^2+3^2+5^2+7^2+....+(2n-1)^2=\dfrac{n(2n-1)(2n+1)}{3}



Using the principle of mathematical induction, prove the following for all n\in N:
1.2+2.2^2+3.2^3+....+n.2^n=(n-1)2^{n+1}+2.



Using the principle of mathematical induction, prove the following for all n\in N:
2+4+6+8+...+2n=n(n+1).



Using the principle of mathematical induction, prove the following for all n\in N:
\dfrac{1}{1.4}+\dfrac{1}{4.7}+\dfrac{1}{7.10}+...+\dfrac{1}{(3n-2)(3n+1)}=\dfrac{n}{(3n+1)}



Using the principle of mathematical induction, prove the following for all n\in N:
(x^{2n}-y^{2n}) is divisible by (x+y).



Using the principle of mathematical induction, prove the following for all n\in N:
3.2^2+3^2.2^3+3^3.2^4+...+3^n.2^{n+1}=\dfrac{12}{5}(6^n-1).



Using the principle of mathematical induction, prove the following for all n\in N:
(3^{2n+2}-8n-9) is divisible by 8.



Using the principle of mathematical induction, prove the following for all n\in N:
n(n+1)(n+2) is a multiple of 6.



Using the principle of mathematical induction, prove the following for all n\in N:
\left\{ (41)^n-(14)^n\right\} is divisible by 27.



Using the principle of mathematical induction, prove the following for all n\in N:
\left(1+\dfrac{1}{1}\right)\left(1+\dfrac{1}{2}\right)\left(1+\dfrac{1}{3}\right)......\left( 1+\dfrac{1}{n}\right)=(n+1).



Using the principle of mathematical induction, prove the following for all n\in N:
\left(1+\dfrac{3}{1}\right)\left(1+\dfrac{5}{4}\right)\left(1+\dfrac{7}{9}\right)......\left\{ 1+\dfrac{(2n+1)}{n^2}\right\}=(n+1)^2.



Using the principle of mathematical induction, prove the following for all n\in N:
(4^n+15n-1) is divisible by 9.



Using the principle of mathematical induction, prove the following for all n\in N:
(2^{3n}-1) is multiple of 7.



Using the principle of mathematical induction, prove the following for all n\in N:
3^n \ge 2^n



Prove that following by using the principle of mathematical induction for all n\in N:
1+3+3^2+....+3^{n-1}=\dfrac{(3^n -1)}{2}



Prove the following by using the principle of mathematical induction for all n\in N:
1.2+2.2^2 +3.2^2+....+n.2^2 =(n-1)2^{n+1}+2



Prove that 3^{2n}+7 is a multiple of 8.



Prove the following by using the principle of mathematical induction for all n\in N:
\dfrac {1}{1.2.3}+\dfrac {1}{2.3.4}+\dfrac {1}{3.4.5}+...+\dfrac {1}{n(n+1)(n+2)}=\dfrac {n(n+3)}{4(n+1)(n+2)}



Prove the following by using the principle of mathematical induction for all n\in N:
\left(1+\dfrac{3}{1}\right)\left(1+\dfrac{5}{4}\right)\left(1+\dfrac{7}{9}\right)......\left(1+\dfrac{(2n+1)}{n^{2}}\right)=(n+1)^{2}



Prove the following by using the principle of mathematical induction for all n\in N:
\dfrac {1}{2}+\dfrac {1}{4}+\dfrac {1}{8}+...+\dfrac {1}{2^n}=1-\dfrac {1}{2^n}



Prove the following by using the principle of mathematical induction for all n\in N:
1.3+3.5+5.7+....+(2n-1)(2n+1)=\dfrac {n(4n^2 +6n-1)}{3}



Prove the following by using the principle of mathematical induction for all n\in N:
\dfrac {1}{2.5}+\dfrac {1}{5.8}+\dfrac {1}{8.11}+...+\dfrac {1}{(3n-1)(3n+2)}=\dfrac {n}{(6n+4)}



Prove the following by using the principle of mathematical induction for all n\in N:
a+ar+ar^2+....+ar^{n-1}=\dfrac {a(r^n -1)}{r-1}



Using the principle of mathematical induction, prove that
1+3+5+7+.....+(2n-1)=n^2, for all n \in N



Prove that following by using the principle of mathematical induction for all n\in N:
1+2+3+.....+n < \dfrac{1}{8}(2n+1)^{2}



Prove the following by using the principle of mathematical induction for all n\in N:
\left(1+\dfrac{1}{1}\right)\left(1+\dfrac{1}{2}\right)\left(1+\dfrac{1}{3}\right)......\left(1+\dfrac{1}{n}\right)=(n+1)



Using the principle of mathematical induction, prove that
1^{2}+2^{2}+3^{2}+\ldots \ldots+n^{2}=\cfrac{n(n+1)(2 n+1)}{6}



Using the principle of mathematical induction, prove that
\dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+....+\dfrac{1}{n(n+1)}=\dfrac{n}{n+1}, for all n\in  N



Prove the following by using the principle of mathematical induction for all n\in N:
1^{2}+3^{2}+5^{2}+.....+(2n-1)^{2}=\dfrac{n(2n-1)(2n+1)}{3}



Using the principle of mathematical induction, prove that
1 \cdot 2+2 \cdot 3+\dots+n(n+1)=\cfrac{n(n+1)(n+2)}{3} for all n \in N



Using the principle of mathematical induction, prove that 
\text { (1) } 1+2+3+\cdots+n=\cfrac{n(n+1)}{2}, for all n\in N



Prove that following by using the principle of mathematical induction for all n\in N:
\dfrac{1}{1.4}+\dfrac{1}{4.7}+\dfrac{1}{7.10}+....+\dfrac{1}{(3n-2)(3n+1)}=\dfrac{n}{(3n+1)}



Prove the following by using the principle of mathematical induction for all n\in N:
\dfrac{1}{3.5}+\dfrac{1}{5.7}+\dfrac{1}{7.9}+....+\dfrac{1}{(2n+1)(2n+3)}=\dfrac{n}{3(2n+3)}



Prove the following by using the principle of mathematical inductions for all n\epsilon N 1 + 3 + 3^{2} + ... + 3^{n - 1} = \dfrac {(3^{n} - 1)}{2}$.



Using principle of mathematical induction, prove that
\cos \alpha \cos 2 \alpha \cos 4\alpha ..... \cos(2^{n-1}\alpha) = \dfrac{\sin 2^n \alpha}{2^n \sin \alpha} for all n \in N.



Using the principle of mathematical induction, prove that
(ab)^n=a^nb^n for all n\in  N



Using the principle of mathematical induction, prove that
7^{2n}-3^{2n} is divisible by 4.



Using the principle of mathematical induction, prove that
(1+x)^n \ge (1+nx) for all natural numbers n, where x > -I.



let   O < A < \pi  for i = 1 , 2  , ...... n  we mathematical induction to prove that 
sinA _1  \, + \,sinA _2  \, \, + \, ....... \, + \, sinA _n   \,
\geq \, n  \, sin \, \left (\dfrac{A_1 \, + \, A_2 \, + \, A_n}{n}   \right )
where n > 1  is a merital number 
p sin x + (1 - p)  sin p \geqsin { px + ( 1 - p) y } 
Where 9 \, \geq p \, \geq \, 1  and 0 \geq x , \, y \, \geq \pi



Prove that n^3 \, + \, 3n^2 \,+\, 5n \, + \, 3 is divisible by 3 for any natural n.



Using  the mathematical induction prove that 
tan^{-1}  (1/3) \, + \,tan^{-1}  (1/7) \, + \,tan^{-1}   [1 / (n^2 \, + \, n \, + \, 10)]
= tan^{-1}   [n / (\, n \, + \, 2)] 



Show that 49^{n}+16n-1 is divisible by 64 for all positive integers n ?



Given (1 + x)^{n} \geq (1 + nx), for all natural number n, where x > -n. What will be the value of n



1+2+3+..............n<\dfrac{1}{8}(2n+1)^2



Using induction method of otherwise, prove that for any non-negative integers m,n,r and k
\sum_{m \, = \, 0}^{k}(n \, - \, m) \, \dfrac{(r \, + \, m)!}{m!} \, = \, \dfrac{(r \, + \, k \, + \, 1)!}{k!}     \left[\dfrac{n}{r \, + \, 1} \, - \, \dfrac{k}{r \, + \, 2}\right]



Use Euclid's division lemma to show that any positive odd integer is of the form 6q+1, or 6q+3,\ 6q+5 where q is some integers.



Using the principle of mathematical induction, prove the following for all n\in N:
\dfrac{1}{2.5}+\dfrac{1}{5.8}+\dfrac{1}{8.11}+...+\dfrac{1}{(3n-1)(3n+2)}=\dfrac{n}{(6n+4)}.



Using the principle of mathematical induction, prove the following for all n\in N:
\dfrac{1}{1.3}+\dfrac{1}{3.5}+\dfrac{1}{5.7}+...+\dfrac{1}{(2n-1)(2n+1)}=\dfrac{n}{(2n+1)}



Using the principle of mathematical induction, prove the following for all n\in N:
1+3+3^2+3^3+....+3^{n-1}=\dfrac{1}{2}(3^n-1).



If A = \left[ \begin{array} { c c } { \cos \theta } & { \sin \theta } \\ { - \sin \theta } & { \cos \theta } \end{array} \right] then show that for all the positive integers n A ^ { n } = \left[ \begin{array} { c c } { \cos n \theta } & { \sin n \theta } \\ { - \sin n \theta } & { \cos n \theta } \end{array} \right]



Show that n(n^2-1)(3n+2) is divisible by 24.



Using the principle of mathematical induction, prove the following for all n\in N:
(x^{2n}-1) is divisible by (x-1), where x\neq 1.



Using the principle of mathematical induction, prove the following for all n\in N:
1+\dfrac{1}{(1+2)}+\dfrac{1}{(1+2+3)}+....+\dfrac{1}{(1+2+3+....+n)}=\dfrac{2n}{(n+1)}.



Prove that ^nC_r + ^{n - 1}C_r + ^{n - 2}C_r + ^{n - 2}C_r +....................+ ^rC_r = ^{n +1}C_{r + 1}



Class 11 Engineering Maths Extra Questions