Mathematical Induction : Sum of N natural number squares

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8 years ago

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Question

Using Mathematical Induction Prove that the sum of first n natural number squares is:

i.e

Using WolframAlpha to find the formula. WolframAlpha is a great mathematical computational tool for Mathematical students.

By Mathematical Induction.

Let’s prove that the base case holds true. We need to show that P(1) is true.

1=1

P(1) is true.

Let’s assume that P(k) hold true. Assume that for some positive k, P(k) holds true, so that

P(k)

We need to show that P(k + 1) holds true.

We have assumed that P(k) is true. Replacing for k terms we get:

Taking (k+1) common we get :

Take hints in the factorization, we need to prove and eventually get (k+1) in terms of k.

expand and rearrange the 2nd term:

Taking k+2 common

Hence P(k + 1) holds true, when P(k) is true, so P(n) is true for all finite natural numbers n.