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# Standard Induction

Induction is a fundamental proof technique that is useful for proving that statements are true for all positive integers n – for example, that 1 + 2 + 3 + ... + n = n(n+1)/2 for positive integers n.

Consider a statement \[S(N): 1+3+5+\cdots+(2N-1)=7+N^2,\] then which of the following is true?

What is the remainder when \({105}^{165}-1\) is divided by 4?

If \(\displaystyle a_n=2^{2^n}+1\) for \(n > 1,\) then what is the last digit of \(a_{451}?\)

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