13.1 Sequences and their notations  (Page 7/15)

$\left(\frac{12}{6}\right)!$

$\frac{12!}{6!}$

$\frac{100!}{99!}$

$a_n=\frac{n!}{n^{\text{2}}}$

$a_n=\frac{3\cdot n!}{4\cdot n!}$

$a_n=\frac{n!}{n^2-n-1}$

$a_n=\frac{100\cdot n}{n(n-1)!}$

$a_n=\frac{{\left(-1\right)}^n}{n}+n$

$a_n=\{\begin{array}{ll}\frac{4+n}{2n}\hfill & \text{if }n\text{ is even}\hfill \\ 3+n\hfill & \text{if }n\text{ is odd}\hfill \end{array}$

$a_1=2,a_n={\left(-a_{n-1}+1\right)}^2$

$a_n=1,a_n=a_{n-1}+8$

$a_n=\frac{\left(n+1\right)!}{\left(n-1\right)!}$

Find the first five terms of the sequence $a_1=\frac{87}{111},a_n=\frac{4}{3}{a}_{n-1}+\frac{12}{37}.$ Use the> Frac feature to give fractional results.

Find the 15 ^th term of the sequence $a_1=625, a_n=0.8a_{n-1}+18.$

Find the first five terms of the sequence $a_1=2, a_n=2^{[(a_n-1)-1]}+1.$

Find the first ten terms of the sequence $a_1=8,a_n=\frac{\left(a_{n-1}+1\right)!}{a_{n-1}!}.$

Find the tenth term of the sequence $a_1=2,a_n=n{a}_{n-1}$

List the first five terms of the sequence $a_n=-\frac{28}{9}n+\frac{5}{3}.$

List the first six terms of the sequence $a_n=\frac{n^3-3.5n^2+ 4.1n-1.5}{2.4n}.$

List the first five terms of the sequence $a_n=\frac{15n\cdot {\left(-2\right)}^{n-1}}{47}$

List the first four terms of the sequence $a_n={5.7}^n+0.275\left(n-1\right)!$

List the first six terms of the sequence $a_n=\frac{n!}{n}.$

Consider the sequence defined by $a_n=-6-8n.$ Is $a_n=-421$ a term in the sequence? Verify the result.

What term in the sequence $a_n=\frac{n^2+4n+4}{2\left(n+2\right)}$ has the value $41?$ Verify the result.

Find a recursive formula for the sequence $1,0,-1,-1,0,1,1,0,-1,-1,0,1,1,\mathrm{...}.$ ( Hint : find a pattern for $a_n$ based on the first two terms.)

Calculate the first eight terms of the sequences $a_n=\frac{\left(n+2\right)!}{\left(n-1\right)!}$ and $b_n=n^3+3n^2+2n,$ and then make a conjecture about the relationship between these two sequences.

Prove the conjecture made in the preceding exercise.