Find the sum of n terms of sereis 1×2×3+2×3×4+3×4×5

Question

Find the sum to n terms of the series \( 1 \times 2 \times 3+2 \times 3 \times 4+3 \times 4 \times 5+\ldots \ldots \)

Answer 1

a_n ​= n(n + 1)(n + 2)

Answer 2

Sum of n terms of series is

S_n = 1 x 2 x 3 + + 2 x 3 x 4 + 3 x 4 x 5 +......+n x (n + 1) x (n + 2)

 = Σn(n + 1)(n + 2)
 = Σn(n² + 3n + 2)

 = Σn (n³ + 2n² + 2n)

 = Σn³ + 3Σn² + 2Σn

 = \((\frac{n(n+1)}2)^2+3\frac{n(n+1)(2n+1)}6+\frac{2n(n+1)}2\) 

 = \(\frac{n(n+1)}{4}\)(n(n + 1) + 2(2n + 1) + 4)

 = \(\frac{n(n+1)}{4}\)(n² + n + 4n + 2 + 4)

 = \(\frac{n(n+1)}{4}\) (n² + 5n + 6)

 = \(\frac{n(n+1)(n+2)(n+3)}4\)