Another geometric sequence

Algebra Level 2

The product of the first five terms of a geometric progression is 1024, and the fourth term is 2017. Find the second term.

If the answer is of the form $\dfrac ab$ , where $a$ and $b$ are positive coprime integers, input your answer as $a+b$ .

1 solution

Chew-Seong Cheong
Mar 7, 2017

Let the third term and the common ration of the geometric progression be $c$ and $r$ respectively, then the product of the first five terms is:

$\begin{aligned} \frac c{r^2} \times \frac cr \times c \times cr \times cr^2 & = 1024 \\ c^5 & = 1024 \\ \implies c & = 4 \end{aligned}$ .

It is given that the fourth term $cr = 2017$ , $\implies r = \dfrac {2017}4$ .

Therefore, the second term $\dfrac cr = \dfrac 4{\frac {2017}4} = \dfrac {16}{2017}$ .

$\implies a+b = 16+2017 = \boxed{2033}$ .