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For the series, S=1+1/((1+3))(1+2)^2+1/(...

For the series, `S=1+1/((1+3))(1+2)^2+1/((1+3+5))(1+2+3)^2+1/((1+3+5+7))(1+2+3+4)^2` +...

A

7th term is 16

B

7th term is 18

C

sum of first 10 terms is `(505)/(4)`

D

sum of first 10 terms is `(405)/(4)`

Text Solution

Verified by Experts

The correct Answer is:
A, C
`:.S=1+(1)/(1+3)(1+2)^(2)+(1)/(1+3+5)(1+2+3)^(2)+"......."`
`T_(n)=(1)/(1+3+5+7+"........"" n terms ")*(1+2+3+"......."" n terms " )^(2)`
`=(1)/([(n)/(2)[2*1+(n-1)*2]])*((n(n+1))/(2))^(2)=((n+1)^(2))/(4)`
(a) `T_(7)=((7+1)^(2))/(4)=(64)/(4)=16`
(b) `S_(10)sum_(n=1)^(10)((n+1)/(2))^(2)=(1)/(4)sum_(n=1)^(10)(n^(2)+2n+1)`
`=(1)/(4)(sum_(n=1)^(10)n^(2)+2sum_(n=1)^(10)n+sum_(n=1)^(10)1)`
`=(1)/(4)((10xx11xx21)/(6)+(2xx10xx11)/(2)+10)`
`=(1)/(4)(385+110+10)=(505)/(4)`
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