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$\left(\frac{12}{6}\right)!$
$\frac{100!}{99!}$
For the following exercises, write the first four terms of the sequence.
${a}_{n}=\frac{n!}{{n}^{\text{2}}}$
First four terms: $1,\frac{1}{2},\frac{2}{3},\frac{3}{2}$
${a}_{n}=\frac{3\cdot n!}{4\cdot n!}$
${a}_{n}=\frac{n!}{{n}^{2}-n-1}$
First four terms: $-1,2,\frac{6}{5},\frac{24}{11}$
${a}_{n}=\frac{100\cdot n}{n(n-1)!}$
For the following exercises, graph the first five terms of the indicated sequence
${a}_{n}=\{\begin{array}{ll}\frac{4+n}{2n}\hfill & \text{if}n\text{iseven}\hfill \\ 3+n\hfill & \text{if}n\text{isodd}\hfill \end{array}$
${a}_{n}=1,\text{}{a}_{n}={a}_{n-1}+8$
For the following exercises, write an explicit formula for the sequence using the first five points shown on the graph.
For the following exercises, write a recursive formula for the sequence using the first five points shown on the graph.
Follow these steps to evaluate a sequence defined recursively using a graphing calculator:
For the following exercises, use the steps above to find the indicated term or terms for the sequence.
Find the first five terms of the sequence ${a}_{1}=\frac{87}{111},\text{}{a}_{n}=\frac{4}{3}{a}_{n-1}+\frac{12}{37}.$ Use the> Frac feature to give fractional results.
First five terms: $\frac{29}{37},\frac{152}{111},\frac{716}{333},\frac{3188}{999},\frac{13724}{2997}$
Find the 15 ^{th} term of the sequence $\text{\hspace{0.17em}}{a}_{1}=625,{a}_{n}=0.8{a}_{n-1}+18.$
Find the first five terms of the sequence $\text{\hspace{0.17em}}{a}_{1}=2,{a}_{n}={2}^{[({a}_{n}-1)-1]}+1.$
First five terms: $2,3,5,17,65537$
Find the first ten terms of the sequence ${a}_{1}=8,\text{}{a}_{n}=\frac{\left({a}_{n-1}+1\right)!}{{a}_{n-1}!}.$
Find the tenth term of the sequence ${a}_{1}=2,\text{}{a}_{n}=n{a}_{n-1}$
${a}_{10}=7,257,600$
Follow these steps to evaluate a finite sequence defined by an explicit formula. Using a TI-84, do the following.
Using a TI-83, do the following.
For the following exercises, use the steps above to find the indicated terms for the sequence. Round to the nearest thousandth when necessary.
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 $\text{\hspace{0.17em}}{a}_{n}=\frac{{n}^{3}-3.5{n}^{2}+4.1n-1.5}{2.4n}.$
First six terms: $0.042,0.146,0.875,2.385,4.708$
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 ${a}_{n}={5.7}^{n}+0.275\left(n-1\right)!$
First four terms: $5.975,32.765,185.743,1057.25,6023.521$
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.
If $\text{\hspace{0.17em}}{a}_{n}=-421\text{\hspace{0.17em}}$ is a term in the sequence, then solving the equation $-421=-6-8n$ for $n$ will yield a non-negative integer. However, if $\text{\hspace{0.17em}}-421=-6-8n,\text{\hspace{0.17em}}$ then $n=51.875$ so ${a}_{n}=-421$ is not a term in the sequence.
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 $1,\text{}0,\text{}-1,\text{}-1,\text{}0,\text{}1,\text{}1,\text{}0,\text{}-1,\text{}-1,\text{}0,\text{}1,\text{}1,\text{}\mathrm{...}\text{}.$ ( Hint : find a pattern for $\text{\hspace{0.17em}}{a}_{n}\text{\hspace{0.17em}}$ based on the first two terms.)
${a}_{1}=1,{a}_{2}=0,{a}_{n}={a}_{n-1}-{a}_{n-2}$
Calculate the first eight terms of the sequences ${a}_{n}=\frac{\left(n+2\right)!}{\left(n-1\right)!}$ and ${b}_{n}={n}^{3}+3{n}^{2}+2n,$ and then make a conjecture about the relationship between these two sequences.
Prove the conjecture made in the preceding exercise.
$\frac{(n+2)!}{(n-1)!}=\frac{(n+2)\xb7(n+1)\xb7(n)\xb7(n-1)\xb7\mathrm{...}\xb73\xb72\xb71}{(n-1)\xb7\mathrm{...}\xb73\xb72\xb71}=n(n+1)(n+2)={n}^{3}+3{n}^{2}+2n$
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