The exercise reads as follows:
The sum of the first 5 terms in a geometric progression is 62. The 5th, 8th and 11th term of this geometric sequence are also the 1st, 2nd and 10th term of an arithmetic sequence. Find the first number of the geometric sequence.
This was a rough translation so my attempt at explaining the exercise-question better is:
Find the number $a$.
GP is geometric progression, AP is arithmetic progression. $x_3,x_4,x_5,x_6,x_7,x_8,x_9$ are the other terms of the arithmetic progression which we know nothing of (and they probably do not matter, not sure).
Any hints? I thought about using the ratio from the geometric progression in two different ways: either starting from $e$ (therefore $d=e/r$ and $h=e*r^3$) or starting from $a$ (and we use $e=a*r^4$). The information about the sum is probably valuable but I might not know how to use it properly.
Thanks to the suggestions, I've made it a bit further:
I've wrote the important terms using $a$ and the geometric progression ratio. We have the following equations:
$q$ is the ratio from the geometric progression, $r$ is the ratio from the arithmetic progression
EDIT: I've managed to find some answers with the help of Ross. The arithmetic progression has the ratio equal to 224, the geometric progression has the ratio 2. The first term is equal to 2. Sadly the book has no answers and I can't double-check, they were discovered by giving values and seeing if it works, assuming all the numbers are integers (which they probably are but the exercise doesn't specify).
If anyone is still willing to help with a solution that doesn't require "trial and error" so to say, feel free.