Question

Find the sum to n terms of each of the series in Exercises 1 to 7 1: 1x2+2x3+3 x4 4 x5+2. 1 x2 x 3+2 x3 4+34 3. 3 x 1 +5 x 22 +7 x3 +... 5. 5+6+72 + 20 Ix2 2x3 3x4 3x8+6x11+9x14 + 6,

Answer 1

The given series is 1×2+2×3+3×4+4×5+...........n terms.

Now, the sum of series is,

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

Solve further.

S n = n( n+1 ) 2 ( 2n+1 3 +1 ) = n( n+1 ) 2 ( 2n+4 3 ) = n( n+1 )( n+2 ) 3

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