Proof by mathematical induction Quiz

1. What is checked in the basis step?

• A. The statement for $n=k$
• B. The statement for $n=k+1$
• C. The statement for the smallest value of $n$, usually $n=1$
• D. The statement for every $n$

2. For $f(n)=3^{2n+1}+1$, what is $f(k+1)-9f(k)$?

• A. $8$
• B. $-8$
• C. $0$
• D. $2$

3. A sequence has $u_{n+1}=3u_n+4$ and $u_k=3^k-2$. What is $u_{k+1}$?

• A. $3^{k+1}-2$
• B. $3^{k+1}+2$
• C. $3^k+2$
• D. $3^{k+1}-6$

4. If $\mathbf{A}^k=\begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix}$ and $\mathbf{A}=\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$, what is the top-right entry of $\mathbf{A}^{k+1}$?

• A. $k$
• B. $2k$
• C. $k^2$
• D. $k+1$

5. For $f(n)=7^n-1$, which of these is $f(k+1)$ in terms of $f(k)$?

• A. $7f(k)+1$
• B. $7f(k)-6$
• C. $7f(k)+6$
• D. $f(k)+7$