Problem Solving Skills

Concept Quizzes

Induction - Problem Solving
Pigeonhole Principle - Problem Solving
Telescoping Series - Sum

Induction - Problem Solving

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

S(5) \text{ is correct.}

S(N) \Rightarrow S(N+1)

S(N) \not\Rightarrow S(N+1)

S(1) \text{ is correct.}

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by Brilliant Staff

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

0 2 3 1

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by Brilliant Staff

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 $a_{50}?$

5^{50}-2^{50}

2^{50}+3^{50}

2^{49}+5^{49}

3^{49}-2^{49}

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by Brilliant Staff

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

5 7 3 4

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by Brilliant Staff

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?

P(1) is true. Both P(1) and P(2) are true. P(2) is true. Both P(1) and P(3) are true.

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by Brilliant Staff

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