13.4 Series and their notations  (Page 3/18)

1. $a_1=8,$ and we are given that $n=11.$

   We can find $r$ by dividing the second term of the series by the first.

   $$r=\frac{-4}{8}=-\frac{1}{2}$$

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

   $$\begin{array}{l}{S}_n=\frac{a_1\left(1-r^n\right)}{1-r}\hfill \\ S_{11}=\frac{8\left(1-{\left(-\frac{1}{2}\right)}^{11}\right)}{1-\left(-\frac{1}{2}\right)}\approx 5.336\hfill \end{array}$$

2. Find $a_1$ by substituting $k=1$ into the given explicit formula.

   $$a_1=3\cdot 2^1=6$$

   We can see from the given explicit formula that $r=2.$ The upper limit of summation is 6, so $n=6.$

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

   $$\begin{array}{l}{S}_n=\frac{a_1(1-r^n)}{1-r}\hfill \\ S_6=\frac{6(1-2^6)}{1-2}=378\hfill \end{array}$$

The problem can be represented by a geometric series with $a_1=26,750\text{;}$ $n=5\text{;}$ and $r=\mathrm{1.016.}$ Substitute values for $a_1\text{,}$ $r\text{,}$ and $n$ into the formula and simplify to find the total amount earned at the end of 5 years.

He will have earned a total of $138,099.03 by the end of 5 years.