Question Type 3: Working with geometric sequences algebraically Bootcamps

The first three terms of a geometric sequence are $a$, $ab$, and $2ab$. Express the common ratio $r$ and determine any condition on $b$.

In a geometric sequence the product of the third and seventh terms is $64$ and the common ratio is $2$. Find the first term.

In a geometric sequence the first term is $8x$ and the common ratio is $\tfrac12$. If the fifth term equals $\tfrac14$, find $x$.

In a geometric sequence $a_2 + a_5 = 45$ and the common ratio is $2$. Find the first term.

The first three terms of a geometric sequence are $5$, $k$, and $20$. Determine all possible values of $k$.

Two geometric sequences satisfy $U_n$ with first term $2$ and ratio $r_1$, and $V_n$ with first term $3$ and ratio $r_2$. If $U_3 = V_3$ and $r_2 = 1/r_1$, find $r_1$.