13.4 Series and their notations  (Page 2/18)

1. We are given $a_1=5$ and $a_n=32.$

   Count the number of terms in the sequence to find $n=10.$

   Substitute values for $a_1,a_n\text{\hspace{0.17em},}$ and $n$ into the formula and simplify.

   $$\begin{array}{l}\begin{array}{l}\hfill \\ S_n=\frac{n(a_1+a_n)}{2}\hfill \end{array}\hfill \\ S_{10}=\frac{10(5+32)}{2}=185\hfill \end{array}$$

2. We are given $a_1=20$ and $a_n=-50.$

   Use the formula for the general term of an arithmetic sequence to find $n.$

   $$\begin{array}{l}{a}_n=a_1+(n-1)d\hfill \\ -50=20+(n-1)(-5)\hfill \\ -70=(n-1)(-5)\hfill \\ 14=n-1\hfill \\ 15=n\hfill \end{array}$$

   Substitute values for $a_1,a_n\text{,}n$ into the formula and simplify.
   

   $$\begin{array}{l}\begin{array}{l}\\ S_n=\frac{n(a_1+a_n)}{2}\end{array}\hfill \\ S_{15}=\frac{15(20-50)}{2}=-225\hfill \end{array}$$

3. To find $a_1,$ substitute $k=1$ into the given explicit formula.

   $\begin{array}{l}{a}_k=3k-8\hfill \\ a_1=3(1)-8=-5\hfill \end{array}$

   We are given that $n=12.$ To find $a_{12},$ substitute $k=12$ into the given explicit formula.

   $$\begin{array}{l}{a}_k=3k-8\hfill \\ a_{12}=3(12)-8=28\hfill \end{array}$$

   Substitute values for $a_1,a_n,$ and $n$ into the formula and simplify.

   $$\begin{array}{l}{S}_n=\frac{n(a_1+a_n)}{2}\hfill \\ S_{12}=\frac{12(-5+28)}{2}=138\hfill \end{array}$$

This problem can be modeled by an arithmetic series with $a_1=\frac{1}{2}$ and $d=\frac{1}{4}.$ We are looking for the total number of miles walked after 8 weeks, so we know that $n=8\text{,}$ and we are looking for $S_8.$ To find $a_8,$ we can use the explicit formula for an arithmetic sequence.

We can now use the formula for arithmetic series.

She will have walked a total of 11 miles.