# The second and the fifth terms of a geometric sequence are 10 and 1250, respectively. Is 31,250 a term of this sequence? If so, which term is it?

Nov 15, 2015

Yes it is a $7 t h$ term; ${a}_{7} = 31250$

#### Explanation:

To solve this task we have to find ${a}_{1}$ and $q$ first:

${a}_{2} = 10$

${a}_{5} = 1250$

${a}_{2} \cdot {q}^{3} = 1250$

$10 {q}^{3} = 1250$

${q}^{3} = 125$

$q = 5$

Now we can calculate ${a}_{1} = {a}_{2} / q = \frac{10}{5} = 2$

Now we have to check if $31250$ is a term of this sequence. To do this we have to solve equation:

$2 \cdot {5}^{n - 1} = 31250$

${5}^{n - 1} = 15625$

${5}^{n - 1} = {5}^{6}$

$n - 1 = 6$

$n = 7$

The number would not be a term in this sequence if the solution of this equation was not a natural number.