Question

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?

Answer 1

Yes it is a `7th` 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*q^3=1250`

`10q^3=1250`

`q^3=125`

`q=5`

Now we can calculate `a_1=a_2/q=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*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.