Sum of power of integers

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SUMS OF POWER OF INTEGERS

If something (actually numbers) is squared, we can think about the area of the square(s).

So take a look at this below:

It is the area of the square! It is amazing! How can we possibly think about this? But someone

did it for us. We must thank them for this.

Now, it is time to analyse it more detail and get the result.

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When we take a look at the square (biggest one) very carefully, we can find the area of white (L

shape but vertically reflected) which has three different parts.

Before going further, take a break to think about this:

That is the sum of a series with a common difference of 1, but just from 1 to th term.

For example,

Now we can back to the square. Without difficulty, we can get the values of the sides of the

squares like this:

is

Therefore,

This mathematical expression can be proofed. The expression shows the relationship between

square and cube. It looks like a magic!

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Now we are going to investigate some of sums of power of integers. They are basic formulas

except one. Let’s find out.

and

A.

First,

We already learned that the relationship above:

B.

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C.

This is the last one. That is not so simple, but we can do it.

How?

How and why? Take a look at the diagram below. The similar concept can be applied.

22

32

42

12

1

1+2

1 + 2 + 3

1 + 2 + 3 + 4 1

4

1 2 3 4

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We can extend this relationship to power of integers. Of course, it is possible. That is a more

generalised form.

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is multiples of

Now we can see the importance of

This is the end of the topic of sum of power of integers.

By Dr Nam