Explanation
{\textbf{Step - 1: Determine the Middle term}}
{\text{Middle term = }}\dfrac{{\text{N}}}{2} + 1 = \dfrac{{2{\text{n}}}}{2} + 1
= \left( {{\text{n + 1}}} \right){\text{th term}}
{\text{The middle term of }}{\left( {{\text{1 + x}}} \right)^{40}}{\text{ is }}\left( {\dfrac{{40}}{2} + 1} \right){\text{ i}}{\text{.e 2}}{{\text{1}}^{th}}{\text{ term in that expansion}}
{{\text{T}}_{21}} = {{\text{ }}^{40}}{{\text{C}}_{20}}{{\text{x}}^{40 - 20}} = {{\text{ }}^{40}}{{\text{C}}_{20}}{{\text{x}}^{20}}
{\text{Therefore the coefficient of the middle term is}}{{\text{ }}^{40}}{{\text{C}}_{20}}
\therefore {{\text{ }}^{40}}{{\text{C}}_{20}} = \dfrac{{40!}}{{20!\left( {40 - 20} \right)!}} = \dfrac{{40!}}{{20!20!}}
{\textbf{Step - 2 : Simplify the term}}{\text{.}}
{\text{Dividing by 20!;and separating the multiples of two in the powers of two we get,}}
\therefore {\text{Middle term = }}\dfrac{{\left( {1.3.5.7.....39} \right){2^{20}}}}{{20!}}
{\textbf{Hence, Correct answer will be }}\dfrac{{\left( {1.3.5.7.....39} \right){2^{20}}}}{{20!}}
(i) (5^{\frac{1}{6}}+7^{\frac{1}{9}})^{1824}
T_{r+1}=^{1824}C_r (5)^{\frac{1824-r}{6}} 7^{\frac{r}{9}}
For integer terms, r should be multiple of 9.
For r=18,36,54,72,.....1818, terms comes as integer.
This is an A.P.
1818=18+(n-1)18
\Rightarrow n=101
Also, for r=0 , we would get an integer
So, total number of terms which gives integer values are 101+1=102.
So, \lambda=102
So, \lambda is divisible by 2,3,17
(ii) (5^{\frac{1}{6}}+2^{\frac{1}{8}})^{1824}
T_{r+1}=^{100}C_r (5)^{\frac{100-r}{6}}2^{\frac{r}{8}}
For rational terms, r should be multiple of 8.
For r=16,40,64,88, terms comes as rational.
So, number of rational terms are 4.
So, \lambda=4
which is divisble by 2.
(iii) (3^{\frac{1}{4}}+4^{\frac{1}{3}})^{99}
T_{r+1}=^{99}C_r (3)^{\frac{99-r}{64}}4^{\frac{r}{3}}
For rational terms, r should be multiple of 3.
For r=3,15,27,.....97, terms comes as rational.
This is an AP
97=3+(n-1)12
\Rightarrow n=8
For r=99 also, there is a rational value
So, number of rational terms are 8+1=9
Now, number of irrational terms = total number of terms -rational number of terms
=99+1-9=91
So, \lambda=91
which is divisble by 7,13.
Hence, option A is the correct answer.
(1+x)^{2n+2}. The middle term will be (\dfrac {N}{2}+1) th term. =\dfrac {2n+2}{2}+1 th term =(n+2)^{th} term Hence coefficient of the middle term will be \:^{2n+2}C_{n+1} =p For (1+x)^{2n+1}. The middle terms will be (\dfrac {N+1}{2}+1) and (\dfrac {N+1}{2}) terms =\dfrac {2n+2}{2}+1 th term and \dfrac {2n+2}{2}^{th} term. =(n+2)^{th} term and (n+1)^{th} term. Hence coefficient of the middle terms will be \:^{2n+1}C_{n+1}=q and \:^{2n+1}C_{n}=r By the properties of binomial coefficients. \:^{2n+1}C_{n+1}+\:^{2n+1}C_{n} =\:^{2n+2}C_{n+1} Hence q+r=p
Please disable the adBlock and continue. Thank you.