# If the fifth and sixth terms of a geometric sequence are 8 and 16, respectively, then what is the first term?

Dec 3, 2016

Since these are consecutive terms, we can use the formula $r = {t}_{n} / {t}_{n - 1}$, where $r$ is the common ratio of the sequence.

$r = {t}_{n} / {t}_{n - 1}$

$r = \frac{16}{8}$

$r = 2$

We now can solve for $a$ in the formula for the nth term of a geometric sequence, ${t}_{n} = a \times {r}^{n - 1}$, if we plug in one of the terms.

Let's use ${t}_{6} = 16$.

$16 = a \times {2}^{6 - 1}$

$16 = a \times {2}^{5}$

$16 = 32 a$

$a = \frac{16}{32}$

$a = \frac{1}{2}$

$\therefore$The first term is $\frac{1}{2}$.

Hopefully this helps!