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Mathematical Induction : Let Pn denote the assertion 'n^2 + 5n + 1 is an even integer.' Prove that Pn+1 is true whenever Pn is true.

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Let Pn denote the assertion "n^2 + 5n + 1 is an even integer."

1. Prove that Pn+1 is true whenever Pn is true.
2. For which n is Pn actually true? What is the moral of this exercise?

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Solution Summary

A proof by mathematical induction is provided.

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Solution:

1) Let's suppose that n^2+5n+1 = even integer and let's denote this statement as P(n)
We will try to prove that P(n+1) = true, that means
(n+1)^2+5(n+1)+1 = even integer
based on the assumption P(n) = true.
We have:
(n+1)^2+5(n+1)+1 = n^2+2n+1+5n+5+1 = (n^2+5n+1)+2n+6 =
= ...

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