Unit 10 test: Sequences, Series and Counting

Find $\displaystyle\lim_{n \to \infty} \frac{4n^2 - 1}{2n^2 + n}$.

Evaluate $\displaystyle\sum_{k=1}^{15} (4k - 1)$.

Find the sum of the infinite series $\displaystyle\sum_{k=0}^{\infty} 12\left(\frac{1}{4}\right)^{k}$.

Evaluate $\displaystyle\sum_{k=1}^{49} \left(\frac{1}{k} - \frac{1}{k+1}\right)$.

For which values of $x$ does the series $\displaystyle\sum_{k=0}^{\infty} (2x)^{k}$ converge?

You want to prove by induction that $2 + 2^2 + 2^3 + \cdots + 2^n = 2^{n+1} - 2$ for all $n \ge 1$. After checking $n = 1$, which of these is the correct inductive step?

In an induction proof that $7^n - 1$ is divisible by $6$, the inductive step rewrites $7^{k+1} - 1 = 7(7^k - 1) + c$. What is $c$?

The inequality $3^n > n^3$ holds for every integer $n$ from some point on. What is the smallest $n_0$ such that $3^n > n^3$ for all integers $n \ge n_0$?

Evaluate $\dbinom{9}{4}$.

Find the coefficient of $x^4$ in the expansion of $(x - 2)^7$.

Find the constant term in the expansion of $\left(2x + \dfrac{1}{x^2}\right)^6$.

Which situation is counted by $\dbinom{10}{3}$ rather than by ${}_{10}P_3$?

Seven people line up for a photo. In how many ways can they line up if two particular friends insist on standing next to each other?

How many distinguishable arrangements are there of the letters in LETTER?

A 3-person committee is chosen from 10 people, including Ava and Raj. How many committees include Ava, Raj or both?