Find the sum of t

In the given question we need to find the sum of the series.

For that, first, we need to find the n^th term of the series so that we can use summation of the series with standard identities and get the required sum.

The series given is (1 × 2 x 3) + (2 × 3 x 4) + (3 × 4 x 5) + … to n terms.

The series can be written as, [(1 x (1 + 1) x (1 + 2)), (2 x (2 + 1) x (2 + 2) … (n x (n + 1) x (n + 2)].

So, n^th term of the series,

a_n = n (n + 1) (n + 2)

a_n = n³ + 3n² + 2n

Now, we need to find the sum of this series, S_n.

Note:

I. Sum of first n natural numbers, 1 + 2 +3+…n,

II. Sum of squares of first n natural numbers, 1² + 2² + 3²+….n²,

III. Sum of cubes of first n natural numbers, 1³ + 2³ + 3³ +…..n³,

IV. Sum of a constant k, N times,

So, for the given series, we need to find,

From, the above identities,

So, Sum of the series,