Given the following concerning an arithmetic series and a geometric series:
The second term of the arithmetic series is the same as the third term of the geometric series. Additionally, the fifth term of the geometric series is the same as the fourteenth term of the arithmetic series.
The first term of the arithmetic series is equal to the second term of the geometric series and three times the first term of the said geometric series.
The sum of the first four terms of the arithmetic series, $SAP_4$ and the sum of the first three terms of the geometric series, $SGP_3$ are related by the formula
$$SAP_4 \;–\; 4\cdot SGP_3 \;+\; 2 \;=\; 0$$
What is the total of the sum of the first nine terms of the arithmetic series and the sum of the first five terms of the geometric series?
Hint: Have you tried writing out the information given? Start by giving names: for example, $a$ and $d$ are the first term and difference of the arithmetic progression, while $g$ and $r$ are the first term and ratio of the geometric progression.
In particular, you should see that the second item gives you some information pretty quickly.
Thus the arithmetic progression is $\{a, a+d, a+2d, \ldots\}$, and the geometric progression is $\{g, gr, gr^2, \ldots\}$.
So, for example, "the second term of the arithmetic progression is the same as the third term of the geometric series" becomes $a+d = g r^2$, and "the first term of the arithmetic progression is the same as the second term of the geometric series" becomes $a = g r$.