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
- 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:
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
Is the property true for the first term?
Let's assume that it is true for
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