3,3×12+5×22 + 7 × 32 +

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Solution

The given series is 3× 1 2 +5× 2 2 +7× 3 2 +....... n th term.

Now, the sum of series is,

S n = ∑ k=1 n a k =2 ∑ k=1 n k 3 + ∑ k=1 n k 2 =2 [ n( n+1 ) 2 ] 2 + n( n+1 )( 2n+1 ) 6 = n 2 ( n+1 ) 2 2 + n( n+1 )( 2n+1 ) 6

Solve further.

S n = n( n+1 ) 2 [ n( n+1 )+ 2n+1 3 ] = n( n+1 ) 2 [ 3 n 2 +3n+2n+1 3 ] = n( n+1 ) 2 [ 3 n 2 +5n+1 3 ] = n( n+1 )( 3 n 2 +5n+1 ) 6

Thus, the sum of series is n( n+1 )( 3 n 2 +5n+1 ) 6 .

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