Explanation
1=1, here n=1⇒1(1+1)2=1 1+2=3, here n=2⇒2(2+1)2=3
1+2+3=6, here n=3⇒3(3+1)2=6
1+2+3+4=10, here n=4⇒4(4+1)2=10
nth term is sum of n terms
Therefore, Tn=n(n+1)2
The top layer has (13×13) balls the layer below it will have (14×14) balls
We have 18 layers
So the total number of balls
N=(13×13)+(14×14)+.......(30×30)N=132+142+.....302
Sum of squares of first n natural numbers is n(n+1)(2n+1)6
∴N= sum of first 30 − sum of first 12
N=30×31×616−12×13×256N=8805
⇒8000<N<9000
\sum _{ r=1 }^{ n }{ \left( { r }^{ 2 }+1 \right) r! }
\Rightarrow \sum_{ r=1 }^{ n }{ ( { r }^{ 2 }+1+2r-2r ) r! }
\Rightarrow \sum_{ r=1 }^{ n }{ ( { ( r+1) }^{ 2 }.r!-2r.r! ) }
\Rightarrow \sum_{ r=1 }^{ n }{ ( { ( r+1) }^{ }.(r+1)!-2r.r! ) }
\Rightarrow \sum_{ r=1 }^{ n }{ [ ( r+2-1 )( r+1 )!-2( ( r+1-1 )r! ) ] }
\Rightarrow \sum_{ r=1 }^{ n }{ [ ( r+2 )( r+1 )!-( r+1 )!-2( ( r+1 )r!-r! ) ] }
\Rightarrow \sum_{ r=1 }^{ n }{ \left [ ( r+2 )!-( r+1 )!-2.( r+1 )!+2.r!) \right] }
T_1=3!-2!- 2.2!+2.1!
T_2=4!-3!-2.3!+2.2!
T_3=5!-4!-2.4!+2.3!
.
T_n=(n+2)!-(n+1)!-2.(n+1)!+2.n!
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Sum=S_n=(n+2)!-2!-2(n+1)!+2.1!
S_n=(n+2)!-2(n+1)!-2+2
S_n=(n+1)![n+2-2]
S_n=n.(n+1)!
Correct answer is C
\cfrac { 5 }{ 7 } =\cfrac { { a }_{ 2 } }{ 2! } +\cfrac { { a }_{ 3 } }{ 3! } +\cfrac { { a }_{ 4 } }{ 4! } +\cfrac { { a }_{ 5 } }{ 5! } +\cfrac { { a }_{ 6 } }{ 6! } +\cfrac { { a }_{ 7 } }{ 7! } \\ \Rightarrow \cfrac { 5 }{ 7 } =\cfrac { 1 }{ 2 } \left( { a }_{ 2 }+\cfrac { 1 }{ 3 } \left( { a }_{ 3 }+\cfrac { 1 }{ 4 } \left( { a }_{ 4 }+......\left( \cfrac { { a }_{ 7 } }{ 7 } \right) \right) \right) \right) \\ \cfrac { 10 }{ 7 } ={ a }_{ 2 }+\cfrac { 1 }{ 3 } \left( { a }_{ 3 }+\cfrac { 1 }{ 4 } \left( { a }_{ 4 }+......\left( \cfrac { { a }_{ 7 } }{ 7 } \right) \right) \right) \\ 1+\cfrac { 3 }{ 7 } ={ a }_{ 2 }+\cfrac { 1 }{ 3 } \left( { a }_{ 3 }+\cfrac { 1 }{ 4 } \left( { a }_{ 4 }+......\left( \cfrac { { a }_{ 7 } }{ 7 } \right) \right) \right)
So, as expression is less than 1
{ a }_{ 2 }=1\\ \cfrac { 3 }{ 7 } =\cfrac { 1 }{ 3 } \left( { a }_{ 3 }+\cfrac { 1 }{ 4 } \left( { a }_{ 4 }+......\cfrac { 1 }{ 5 } \left( { a }_{ 5 }+.....\cfrac { { a }_{ 7 } }{ 7 } \right) \right) \right) \\ 1+\cfrac { 2 }{ 7 } ={ a }_{ 3 }+\cfrac { 1 }{ 4 } \left( { a }_{ 4 }+......\cfrac { 1 }{ 5 } \left( { a }_{ 5 }+\cfrac { 1 }{ 6 } \left( { a }_{ 6 }+\cfrac { { a }_{ 7 } }{ 7 } \right) \right) \right) \\ { a }_{ 3 }=1
Similarly
{ a }_{ 4 }=1\quad \quad { a }_{ 7 }=2\\ { a }_{ 5 }=0\\ { a }_{ 6 }=4
So, Sum { a }_{ 7 }-1+1+1+4+2\\ =9
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