Proof by mathematical induction Quiz

1. In proving $\sum_{r=1}^{n}r^3=\frac14n^2(n+1)^2$ by induction, what is added to $\frac14k^2(k+1)^2$ in the inductive step?

• A. $k^3$
• B. $(k+1)^2$
• C. $(k+1)^3$
• D. $\frac14(k+1)^3$

2. For $\mathrm{f}(n)=3^{2n}+11$, what is $\mathrm{f}(k+1)-\mathrm{f}(k)$?

• A. $8\times3^{2k}$
• B. $3^{2k}$
• C. $9\times3^{2k}+11$
• D. $6\times3^{2k}$

3. Which is a complete final conclusion?

• A. It is true for $n=k+1$.
• B. It is true for $n=1$ so it is true for all $n$.
• C. Assuming true for $n=k$, the result holds.
• D. True for $n=1$; if true for $n=k$ then true for $n=k+1$; so true for all positive integers $n$.

4. Claim $\mathbf{M}^n=\begin{pmatrix}1&n\\0&1\end{pmatrix}$ for $\mathbf{M}=\begin{pmatrix}1&1\\0&1\end{pmatrix}$. What is the top-right entry of $\mathbf{M}^k\mathbf{M}$?

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

5. What is the value of $3^{2n}+11$ when $n=1$, and is it divisible by 4?

• A. $14$, no
• B. $12$, yes
• C. $20$, yes
• D. $29$, no