13.4 Series and their notations  (Page 5/18)

1. There is not a constant ratio; the series is not geometric.

2. There is a constant ratio; the series is geometric. $a_1=248.6$ and $r=\frac{99.44}{248.6}=0.4,$ so the sum exists. Substitute $a_1=248.6$ and $r=0.4$ into the formula and simplify to find the sum:

   $$\begin{array}{l}S=\frac{a_1}{1-r}\hfill \\ S=\frac{248.6}{1-0.4}=414.\overline{3}\hfill \end{array}$$

3. The formula is exponential, so the series is geometric with $r=–\frac{1}{3}.$ Find $a_1$ by substituting $k=1$ into the given explicit formula:

   $$a_1=4\text{,}374\cdot {(–\frac{1}{3})}^{1–1}=4\text{,}374$$

   Substitute $a_1=4\text{,}374$ and $r=-\frac{1}{3}$ into the formula, and simplify to find the sum:

   $$\begin{array}{l}S=\frac{a_1}{1-r}\hfill \\ S=\frac{4\text{,}374}{1-(-\frac{1}{3})}=3\text{,}280.5\hfill \end{array}$$
4. The formula is exponential, so the series is geometric, but $r>1.$ The sum does not exist.

We notice the repeating decimal $0.\overline{3}=\mathrm{0.333...}$ so we can rewrite the repeating decimal as a sum of terms.

Looking for a pattern, we rewrite the sum, noticing that we see the first term multiplied to 0.1 in the second term, and the second term multiplied to 0.1 in the third term.

Notice the pattern; we multiply each consecutive term by a common ratio of 0.1 starting with the first term of 0.3. So, substituting into our formula for an infinite geometric sum, we have