Extension Test: Geometric Sequences and Series Solutions

1. Determine the common ratio.

$r=\frac{–320}{1600}=–0.2$

2. Find the seventh term in the sequence.

$$\begin{gathered} a_1=1600 \\ r=–0.2 \\ n=7 \\ {a}_7=\left(a_1\right){\left(–0.2\right)}^{7–1} \\ {a}_7=\left(1600\right){\left(–0.2\right)}^6 \\ {a}_7=0.1024 \end{gathered}$$

3. Demi decides she needs to have 5 rest days during the month-long marathon. What is the starting distance on day one? (Round to the nearest hundredth of a km.)

$\begin{array}{ccc}{S}_{25}=42& & S_{25}=\frac{a_1\left(1–1.1^{25}\right)}{1–1.1}\\ r=1.1& & \left(–0.1\right)42=\frac{a_1\left(1–1.1^{25}\right)}{\cancel{–0.1}}\cancel{\left(–0.1\right)}\\ n=25& & \frac{–4.2}{\left(1–1.1^{25}\right)}=\frac{a_1\cancel{\left(1–1.1^{25}\right)}}{\cancel{\left(1–1.1^{25}\right)}}\\ & & a_1=\frac{–4.2}{\left(1–1.1^{25}\right)}=0.4270\\ & & a_1=0.43\end{array}$ 

Demi will run 0.43 km on day one.

4. What distances will Demi run for the first 3 days?

$$\begin{gathered} a_1=0.43 \\ {a}_{n+1}=a_1{r}^{n–1}=\left(0.43\right){\left(1.1\right)}^{n–1} \\ {a}_2=\left(0.43\right){\left(1.1\right)}^{2–1}=0.473 \\ {a}_3=\left(0.43\right){\left(1.1\right)}^{3–1}=0.5203 \end{gathered}$$ 

Day 1: 0.43 km
Day 2: 0.47 km
Day 3: 0.52 km

5. What is the total distance Demi has completed after 10 days of running?

$$\begin{gathered} n=10 \\ {S}_{10}=\frac{0.43–0.43{\left(1.1\right)}^{10}}{1–1.1} \\ {S}_{10}=6.8531 \end{gathered}$$

After 10 days, Demi ran a total of 6.85 km.