# If the third term of a geometric sequence is 36 and the eighth term is 8748, how do you find the first term?

Nov 8, 2015

${A}_{1} = 4$

#### Explanation:

${A}_{n} = {A}_{1} {r}^{n - 1}$

$\implies {A}_{3} = {A}_{1} {r}^{2}$

$\implies 36 = {A}_{1} {r}^{2}$

${A}_{8} = {A}_{1} {r}^{7}$

$\implies 8748 = {A}_{1} {r}^{7}$

If we divide the second equation by the second

$\frac{8748}{36} = \frac{{A}_{1} {r}^{7}}{{A}_{1} {r}^{2}}$

$\implies 243 = {r}^{5}$

$\implies r = 3$

Let's use the first equation to find ${A}_{1}$

${A}_{3} = {A}_{1} {r}^{2}$

$\implies 36 = {A}_{1} \cdot {3}^{2}$

$\implies 36 = 9 {A}_{1}$

$\implies {A}_{1} = 4$