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What are the main differences between using a recursive formula and using an explicit formula to describe an arithmetic sequence?
Describe how linear functions and arithmetic sequences are similar. How are they different?
Both arithmetic sequences and linear functions have a constant rate of change. They are different because their domains are not the same; linear functions are defined for all real numbers, and arithmetic sequences are defined for natural numbers or a subset of the natural numbers.
For the following exercises, find the common difference for the arithmetic sequence provided.
$\{5,11,17,23,29,\mathrm{...}\}$
$\left\{0,\frac{1}{2},1,\frac{3}{2},2,\mathrm{...}\right\}$
The common difference is $\frac{1}{2}$
For the following exercises, determine whether the sequence is arithmetic. If so find the common difference.
$\{11.4,9.3,7.2,5.1,3,\mathrm{...}\}$
$\{4,16,64,256,1024,\mathrm{...}\}$
The sequence is not arithmetic because $16-4\ne 64-16.$
For the following exercises, write the first five terms of the arithmetic sequence given the first term and common difference.
${a}_{1}=\mathrm{-25}$ , $d=\mathrm{-9}$
${a}_{1}=0$ , $d=\frac{2}{3}$
$$0,\text{\hspace{0.17em}}\frac{2}{3},\text{\hspace{0.17em}}\frac{4}{3},\text{\hspace{0.17em}}2,\text{\hspace{0.17em}}\frac{8}{3}$$
For the following exercises, write the first five terms of the arithmetic series given two terms.
${a}_{1}=17,\text{\hspace{0.17em}}{a}_{7}=-31$
${a}_{13}=-60,\text{\hspace{0.17em}}{a}_{33}=-160$
$0,-5,-10,-15,-20$
For the following exercises, find the specified term for the arithmetic sequence given the first term and common difference.
First term is 3, common difference is 4, find the 5 ^{th} term.
First term is 4, common difference is 5, find the 4 ^{th} term.
${a}_{4}=19$
First term is 5, common difference is 6, find the 8 ^{th} term.
First term is 6, common difference is 7, find the 6 ^{th} term.
${a}_{6}=41$
First term is 7, common difference is 8, find the 7 ^{th} term.
For the following exercises, find the first term given two terms from an arithmetic sequence.
Find the first term or ${a}_{1}$ of an arithmetic sequence if ${a}_{6}=12$ and ${a}_{14}=28.$
${a}_{1}=2$
Find the first term or ${a}_{1}$ of an arithmetic sequence if ${a}_{7}=21$ and ${a}_{15}=42.\text{\hspace{0.17em}}$
Find the first term or ${a}_{1}$ of an arithmetic sequence if ${a}_{8}=40$ and ${a}_{23}=115.$
${a}_{1}=5$
Find the first term or ${a}_{1}$ of an arithmetic sequence if ${a}_{9}=54$ and ${a}_{17}=102.$
Find the first term or ${a}_{1}$ of an arithmetic sequence if ${a}_{11}=11$ and ${a}_{21}=16.$
${a}_{1}=6$
For the following exercises, find the specified term given two terms from an arithmetic sequence.
${a}_{1}=33\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{a}_{7}=-15.$ Find $\text{\hspace{0.17em}}{a}_{4}.$
${a}_{3}=-17.1\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{a}_{10}=-\mathrm{15.7.}$ Find ${a}_{21}.$
${a}_{21}=-13.5$
For the following exercises, use the recursive formula to write the first five terms of the arithmetic sequence.
${a}_{1}=39;\text{}{a}_{n}={a}_{n-1}-3$
${a}_{1}=-19;\text{}{a}_{n}={a}_{n-1}-1.4$
$-19,-20.4,-21.8,-23.2,-24.6$
For the following exercises, write a recursive formula for each arithmetic sequence.
${a}_{n}=\left\{40,60,80,\mathrm{...}\right\}$
${a}_{n}=\{17,26,35,\mathrm{...}\}$
$\begin{array}{ll}{a}_{1}=17;{a}_{n}={a}_{n-1}+9\hfill & n\ge 2\hfill \end{array}$
${a}_{n}=\{-1,2,5,\mathrm{...}\}$
${a}_{n}=\{12,17,22,\mathrm{...}\}$
$\begin{array}{ll}{a}_{1}=12;{a}_{n}={a}_{n-1}+5\hfill & n\ge 2\hfill \end{array}$
${a}_{n}=\{-15,-7,1,\mathrm{...}\}$
${a}_{n}=\{8.9,10.3,11.7,\mathrm{...}\}$
$\begin{array}{ll}{a}_{1}=8.9;{a}_{n}={a}_{n-1}+1.4\hfill & n\ge 2\hfill \end{array}$
${a}_{n}=\{-0.52,-1.02,-1.52,\mathrm{...}\}$
${a}_{n}=\left\{\frac{1}{5},\frac{9}{20},\frac{7}{10},\mathrm{...}\right\}$
$\begin{array}{ll}{a}_{1}=\frac{1}{5};{a}_{n}={a}_{n-1}+\frac{1}{4}\hfill & n\ge 2\hfill \end{array}$
${a}_{n}=\left\{-\frac{1}{2},-\frac{5}{4},-2,\mathrm{...}\right\}$
${a}_{n}=\left\{\frac{1}{6},-\frac{11}{12},-2,\mathrm{...}\right\}$
$\begin{array}{ll}{}_{1}=\frac{1}{6};{a}_{n}={a}_{n-1}-\frac{13}{12}\hfill & n\ge 2\hfill \end{array}$
For the following exercises, write a recursive formula for the given arithmetic sequence, and then find the specified term.
${a}_{n}=\left\{7\text{,}4\text{,}1\text{,}\mathrm{...}\right\};\text{\hspace{0.17em}}$ Find the 17 ^{th} term.
${a}_{n}=\left\{4\text{,}11\text{,}18\text{,}\mathrm{...}\right\};\text{\hspace{0.17em}}$ Find the 14 ^{th} term.
${a}_{1}=4;\text{}{a}_{n}={a}_{n-1}+7;\text{}{a}_{14}=95$
${a}_{n}=\left\{2\text{,}6\text{,}10\text{,}\mathrm{...}\right\};\text{\hspace{0.17em}}$ Find the 12 ^{th} term.
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