Question

9. n2+2

Answer 1

The given n th term of the series is,

a n = n 2 + 2 n

So, the sum of series is,

S n = ∑ k=1 n a k = ∑ k=1 n ( k 2 + 2 k ) = ∑ k=1 n k 2 + ∑ k=1 n 2 k

Assume,

∑ k=1 n 2 k = 2 1 + 2 2 + 2 3 +…

This shows that the above series is a G.P. with first term as 2 and common ratio also as 2. So, it can be written as,

∑ k=1 n 2 k = ( 2 )[ ( 2 ) n −1 ] 2−1 =( 2 )[ ( 2 ) n −1 ]

Sum of series becomes,

S n = ∑ k=1 n k 2 +2( 2 n −1 ) = n( n+1 )( 2n+1 ) 6 +2( 2 n −1 )

Therefore, the sum of n terms of series is n( n+1 )( 2n+1 ) 6 +2( 2 n −1 ).