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# Induction - Problem Solving

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?

Consider a sequence $$\{a_n\}$$ with $$a_1=5$$ and $$a_2=13.$$ If the sequence satisfies $a_{n+2}=5a_{n+1}-6a_n$ for all positive integers $$n,$$ what is $$a_{50}?$$

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

Let $$P(n)$$ be a statement involving a positive integer $$n.$$ $$P(n+2)$$ is true if $$P(n)$$ or $$P(n+1)$$ is true. Then what is the sufficient condition for the statement $$P(n)$$ to be true for all positive integers?

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