Some Special Sequences, Sum of the Powers of the First N Natural Numbers

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Sum of the Powers of the First N Natural Numbers:

(A) the Series of First N Natural Numbers:

Let

This is an arithmetic series whose the first term is , the common difference is and the number of terms is

i.e.

We can write

(B) Determine the Sum of the Squares of the First N Natural Numbers:

Let

Consider the identity:

By giving the values for in the above identity, we have

……… ……….

……… ……….

Adding these we get

Or

Or

i.e.

(C) Determine the Sum of the Cubes of the First N Natural Numbers:

  • Here

  • Consider the identity:

  • Putting successively for we have

……… ……….

……… ……….

Adding these, we get

i.e.

or

Example:

Find the sum of first n terms of the series

Solution:

Let

The term of the series

term ofterm of

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