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