Answer :

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 × 4) + (2 × 3 × 7) + (3 × 4 × 10) + … to n terms.

The series can be written as, [(1 x (1 + 1) x (3 x 1 + 1)), (2 x (2 + 1) x (3 x 2 + 1))… (n x (n + 1) x (3 x n + 1)].

So, n^{th} term of the series,

a_{n} = n (n + 1) (3n + 1)

a_{n} = 3n^{3} + 4n^{2} + n

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, 1^{2} + 2^{2} + 3^{2}+….n^{2},

III. Sum of cubes of first n natural numbers, 1^{3} + 2^{3} + 3^{3} +…..n^{3},

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,

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