08 Sequences and Recurrences Online Quiz Questions

An arithmetic sequence has $a_0=3$ and common difference $d=4$. What is $\sum_{k=0}^{4}a_k$?

For the homogeneous recurrence $a_n=4a_{n-1}-3a_{n-2}$, which is the correct general solution?

Let $s_n$ count lists with entries from $\{1,2,3\}$ whose entries sum to $n$, with $s_0=1$ and $s_n=0$ for $n<0$. What is $s_4$?

Select all statements that are correct about recurrence relations.

Select all correct statements about solving nonhomogeneous linear recurrences with constant coefficients.

True or false: A recurrence relation without sufficient initial conditions generally fails to determine a unique sequence.

True or false: The recurrence $T(n)=2T(n/2)+cn$ models merge sort and has running time $O(n\log n)$.

A sequence is defined by $a_0=2$ and $a_n=a_{n-1}+3$ for $n\ge 1$. Enter the exact value of $a_5$.

What is the order of the recurrence $a_n=5a_{n-1}-2a_{n-2}+7$? Enter the exact integer.

Construct a recurrence for the number of tilings of a $1\times n$ board using squares of length 1 and dominoes of length 2. Define the counted quantity, justify the recurrence by considering the final tile, and give the necessary initial conditions.

What is the sum $\sum_{k=0}^{3}2\cdot 3^k$?

An arithmetic sequence has $a_0=4$ and common difference $d=3$. What is $\sum_{k=0}^{4}a_k$?

What is the order of the recurrence $a_n=4a_{n-1}-a_{n-2}$?

Evaluate the infinite geometric series $\sum_{k=0}^{\infty}6\left(\frac{1}{3}\right)^k$.

What is the value of $\sum_{k=1}^{4}\left(\frac{1}{k}-\frac{1}{k+1}\right)$?

True or false: A recurrence relation without sufficient initial conditions generally determines a unique sequence.

Given $a_0=1$ and $a_n=2a_{n-1}+3$ for $n\ge 1$, enter the value of $a_3$.

In solving a nonhomogeneous linear recurrence, what is the standard name for the additional solution combined with the homogeneous solution to obtain the general solution?