Question

5. 52 + 62 7.202

Answer 1

The series is given as,

5 2 + 6 2 + 7 2 +⋯+ 20 2

The n th term of the given series is,

a n = ( n+4 ) 2 = n 2 +16+8n

The sum of the given series is,

S n = ∑ k=1 n a k = ∑ k=1 n ( k 2 +8k+16 ) = ∑ k=1 n k 2 + ∑ k=1 n 8k + ∑ k=1 n 16 = k( k+1 )( 2k+1 ) 6 + 8k( k+1 ) 2 +16k

Since, the given series has 16 terms, then, substitute k=16 in the sum of series,

S 16 = 16( 16+1 )( 2( 16 )+1 ) 6 + 8( 16 )( 16+1 ) 2 +16( 16 ) = 16( 17 )( 33 ) 6 + 2176 2 +256 = 8976 6 + 2176 2 +256 =2840

Therefore, the sum to the given series is 2840.