Question Type 3: Working with geometric sequences algebraically Exercises

The $n^{\text{th}}$ term of a geometric sequence is given by $T_n=5\cdot3^{n-1}$. Calculate the sum of the first $6$ terms.

A geometric sequence has first term $3x$ and common ratio $2$. Find its $6^{\text{th}}$ term in terms of $x$.

A geometric sequence has first term $k$ and third term $9k$. Find the common ratio $r$.

Given the terms of a geometric sequence are $a$, $ab$, and $2ab$, find the common ratio $r$ in terms of $a$ and $b$. Assume $a\neq0$ and $b\neq0$.

Two geometric sequences are defined by $g_n=2\cdot3^{n-1}$ and $h_n=8\cdot r^{n-1}$. If they have the same fourth term, find $r$.

In a geometric sequence, the second term is $4$ and the fifth term is $32$. Find the first term and the common ratio.

In a geometric sequence, the third term is $12$ and the sixth term is $96$. Determine the common ratio $r$.

A geometric sequence has first term $T_1=2$ and second term $T_2=2p$. If the fourth term $T_4=18$, find $p$.

A geometric sequence has first term $2$ and the sum of its first $4$ terms is $30$. Find the common ratio $r$.

A geometric sequence has $T_1=a$, $T_2=b$, and $T_4=8b$. Express the common ratio $r$ in terms of $b$, then find $a$ in terms of $b$.

The product of the first three terms of a geometric sequence is $216$, and the second term is $6$. Find the common ratio $r$ and the first term.

A geometric sequence has general term $T_n=Ar^{n}$. If $T_1=5$ and its sum to infinity is $15$, find $r$ and $A$.