Quantitative Finance

Problem Solving Skills

Induction - Problem Solving


Consider a statement S(N):1+3+5++(2N1)=7+N2,S(N): 1+3+5+\cdots+(2N-1)=7+N^2, then which of the following is true?

What is the remainder when 1051651{105}^{165}-1 is divided by 4?

Consider a sequence {an}\{a_n\} with a1=5a_1=5 and a2=13.a_2=13. If the sequence satisfies an+2=5an+16ana_{n+2}=5a_{n+1}-6a_n for all positive integers n,n, what is a50?a_{50}?

If an=22n+1\displaystyle a_n=2^{2^n}+1 for n>1,n > 1, then what is the last digit of a451?a_{451}?

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


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