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

Given a geometric series with $S_4 = 15$ and $S_6 = 63$, find the first term and common ratio.

The sum of the first $n$ terms of a series is given by $S_n = n^2 + 5n$. Find $n$ if $S_n = 84$.

A geometric series has common ratio $r = 2$ and sum of the first 4 terms $S_4 = 15$. Find the first term and $S_6$.

A geometric series has sum to infinity $S_\infty =16$ and sum of the first two terms $S_2 =12$. Find the possible values of the first term $a$ and the common ratio $r$.

Given an arithmetic series with $S_4 = 22$ and $S_7 = 70$, find the first term and common difference.

The sum of the first $n$ terms of an arithmetic series is given by $S_n = 6n - n^2.$ Find its first term and common difference.

The sum of the first $n$ terms of a series is given by $S_n = 3n^2 + n.$ Assuming the series is arithmetic, find its first term and common difference.

Given a geometric series with positive terms such that $S_2 = 5$ and $S_6 = 105$, find the first term and common ratio.

Given an arithmetic series with $S_8 = 208$ and $S_{12} = 456$, find the first term and common difference.

Given an arithmetic series with $S_4 = 26$ and $S_{10} = 155$, find $S_{15}$.

Given a geometric series with $S_3 = 7$ and $S_5 = 31$, find the common ratio and the sum of the first 7 terms.

Given an arithmetic series with $S_7 = 77$ and $S_{15} = 345$, find $S_8$.