Question

2 х 3 + 2 х 3 х 4 + 3 х 4 х 5 +

Answer 1

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

Now, the sum of series is,

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

Solve further.

S n = n( n+1 ) 2 [ n( n+1 ) 2 +2n+1+2 ] = n( n+1 ) 2 [ n 2 +n+4n+6 2 ] = n( n+1 ) 2 [ n 2 +5n+6 2 ] = n( n+1 )[ n( n+2 )+3( n+2 ) ] 4

On solving further, we get

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

Thus, the sum of series is n( n+1 )( n+2 )( n+3 ) 4 .