# Question #81d37

##### 2 Answers

This is a classical case in which you can prove something by induction. The idea is the following: think of a river with very strong current that you must cross. You can't swim, but there are several stones, and you can jump from one to the other. If you find a first (solid!) stone, and you learn how to jump from a stone to the next, then you can cross the river!

In formulas, a proof by induction works as follows:

- First of all, you must prove that the formula holds for a basic case (which is usually
#n=0# or#n=1# ) - You must prove that, if the formula holds for a generic value
#n# , then it also works for#n+1# .

By doing so, you can work -indeed- by induction, and step from a number to its successor, and so cover the entire range of natural numbers: the statement it's true for

So, let's begin with the basic case

Of course it is, since the sum means simply 1^3, which is one, and the fraction on the right side is also 1.

So now, let's assume that the sum of the first

We want this to be equal to

which is exactly what we wanted.

First of all let's try to understand *why* it works.

We know that

and this is because we have

Now we can see our sum:

Let's proceed step by step:

so

so

so

...

so **has to be**

**But this is not a demonstration**.

This is **only** an *idea*.

We have to demonstrate it with the induction rule, that says:

If a property is true for the first value, we can assume that it is true for the

So:

Is the property true for the first term?

**YES**

Let's assume that it is true for

**hypotesis**.

Let's demonstrate that it is true also for

that is perfectly the formula that we have to demonstrate in which at the place of