DEAMER 2.0 INTRO TO SEQUENCES/SERIES (4/14/2026)

Write an explicit formula for the $n^{th}$ term of the sequence $1, \frac{1}{2}, \frac{1}{6}, \frac{1}{24}, \frac{1}{120}$. (Assume that $n$ begins with 1.)

Write an explicit formula for the $n^{th}$ term of the sequence $1, \frac{2^2}{2}, \frac{2^3}{6}, \frac{2^4}{24}$. (Assume that $n$ begins with 1.)

Write an explicit formula for the $n^{th}$ term of the sequence $2, 1, \frac{4}{5}, \frac{2}{7}, \frac{3}{3}$. (Assume that $n$ begins with 1.)
Hint: rewrite the fractions.

Write an explicit formula for the $n^{th}$ term of the sequence where $a_1 = 25$ and $a_{n+1} = a_n - 5$. (Assume that $n$ begins with 1.)

Write an explicit formula for the $n^{th}$ term of the sequence where $a_1 = 6$ and $a_{n+1} = a_n + 2$. (Assume that $n$ begins with 1.)

Write an explicit formula for the $n^{th}$ term of the sequence where $a_1 = 6$ and $a_{n+1} = \frac{1}{3} a_n$. (Assume that $n$ begins with 1.)

Write the first five terms of the sequence $a_n = \frac{3^n}{n!}$. (Assume that $n$ begins with 0.)

Write the first five terms of the sequence $a_n = \frac{1}{(n+1)!}$. (Assume that $n$ begins with 0.)

Write the first five terms of the sequence $a_n = \frac{(-1)^{2n+1}}{(2n+1)!}$. (Assume that $n$ begins with 0.)

Simplify the factorial expression $\frac{5!}{8!}$.

Simplify the factorial expression $\frac{212!}{209!}$.

Simplify the factorial expression $\frac{115!}{116!}$.

Simplify the factorial expression $\frac{(n+1)!}{n!}$.

Simplify the factorial expression $\frac{(n+1)!}{(n+3)!}$.

Simplify the factorial expression $\frac{(n+4)!}{(n+2)!}$.