# 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