Second-order recurrence relations Quiz

1. What are the roots of the auxiliary equation for $u_{n+2}-5u_{n+1}+6u_n=0$?

• A. $m=1$ and $m=6$
• B. $m=2$ and $m=3$
• C. $m=-2$ and $m=-3$
• D. $m=5$ and $m=6$

2. The auxiliary equation of a recurrence relation has a repeated root $m=2$. What is the complementary function?

• A. $A\times2^n+B\times2^n$
• B. $A\times2^n+B\times4^n$
• C. $A\times2^n+B(-2)^n$
• D. $(A+Bn)2^n$

3. Find the constant particular solution of $u_{n+2}-5u_{n+1}+6u_n=4$.

• A. $2$
• B. $4$
• C. $-2$
• D. $\frac13$

4. For $u_{n+2}=5u_{n+1}-6u_n$ with $u_1=1$ and $u_2=5$, what is $u_3$?

• A. $25$
• B. $31$
• C. $19$
• D. $11$

5. Which must be assumed in an inductive step for a second-order recurrence relation?

• A. true for $n=k$ only
• B. true for $n=k$ and $n=k+1$
• C. true for $n=k+2$
• D. true for $n=1$ only