S_{5}= a(q^{5}-1)/q-1 =242
S_{10}-S_{5}= a(q^{10}-1)/q-1 - a(q^{5}-1)/q-1 = 58806
Tip: I think dividing the equation on each others would be easier to solve
S_{5}= a(q^{5}-1)/q-1 =242
S_{10}-S_{5}= a(q^{10}-1)/q-1 - a(q^{5}-1)/q-1 = 58806
Tip: I think dividing the equation on each others would be easier to solve
Sounds like a good idea; a lot of things cancel.
The left sides of both equations have the common factor $\displaystyle \frac{a(q^5-1)}{q-1}$. Divide the right-hand sides by this factor and the division you suggested will cancel them.
Hint: $\displaystyle q^{10}-1 = (q^5-1) (q^5 + 1)$