The mathematical induction is used to proof sum, or row building roles. The idea of it is to apply a building role P(n) to the element n+1 what is P(n+1) and compare it to P(n) extended by the element n+1 and if both are equal and the role is valid for the first element x < n P(x) as well, then the building role is correct.
If we have the sample
That leads to the assumption
To proof that:
and
Now P(n) plus the element n+1
and
And with i = 1
The role is proofed
Or for the sum of the squares as another sample:
Proof:
What proofs the role as correct.
Another example: Pascal's triangle (here for n = 0…4 and k ≤ n):
For the binominal coefficients (a+b)k we get the following elements:
This triangle form is called Pascal’s triangle.
It has the building role for the factors of each term:
with
If we look at one element, it is always built of the upper 2 elements right above it:
And the building role for this element between the two upper is
For 1 < k < n
For the proof:
and
What proofs this role as correct as well