Question Type 6: Finding common ratio or sum of first n terms given the values of other sums Exercises

The sum of the first 5 terms of a geometric series is 31 and the sum of the first 3 terms is 7. Given that the common ratio $r$ is an integer, find the first term $a$ and common ratio $r$.

The sum of the first 4 terms of a geometric series is 15 and the sum of the first 7 terms is 127. Given that the common ratio $r$ is an integer, find $a$ and $r$.

In a geometric series, $S_5=31$ and the sum of the 6th through 10th terms is 992. Find $u_1$ and $r$.

In a geometric series, $S_8=255$ and $S_5=31$. Find $S_6$.

The sum of the first 4 terms of a geometric series is 15, and the sum of the 2nd through 5th terms is 30. Determine $a$ and $r$.

Given $S_4=15$ and $S_6=63$ for a geometric series, find the sum of the first 10 terms $S_{10}$.