# How do you find the common ratio, r, for the geometric sequence that has a1 =100 and a8 =50?

Feb 27, 2016

$r = {\left(\frac{1}{2}\right)}^{\frac{1}{7}} \approx 0.9057$

#### Explanation:

The key to solving this problem is in recognizing what the terms of a geometric sequence look like. If we write a geometric sequence in its most general form, we have:

${a}_{1} = {a}_{1}$
${a}_{2} = {a}_{1} r$
${a}_{3} = {a}_{1} {r}^{2}$
...
${a}_{n} = {a}_{1} {r}^{n - 1}$

In this case ${a}_{8} = 50$. But, using ${a}_{1} = 100$, we have

$50 = {a}_{8} = {a}_{1} {r}^{8 - 1} = 100 {r}^{7}$

$\implies {r}^{7} = \frac{50}{100} = \frac{1}{2}$

$\therefore r = {\left(\frac{1}{2}\right)}^{\frac{1}{7}} \approx 0.9057$