The nth term u_n of a geometric sequence is given by u_n = 3(4)^(n+1), n in ZZ^+. What is the common ratio r?

I supposed $r$ might be $4$ but the $n$th term of a geometric sequence is normally given by ${u}_{n} = {u}_{1} \cdot {r}^{n - 1}$, but in the question $4$ has a power of $n + 1$ so I'm confused by $n + 1$ and $n - 1$

Nov 25, 2017

$4.$

Explanation:

The Common Ratio $r$ of a Geometric Sequence

$\left\{{u}_{n} = {u}_{1} \cdot {r}^{n - 1} : n \in {\mathbb{Z}}^{+}\right\}$ is given by,

$r = {u}_{n + 1} \div {u}_{n} \ldots \ldots \ldots \ldots . \left(\ast\right) .$

Since, ${u}_{n} = 3 \cdot {4}^{n + 1} ,$ we have, by $\left(\ast\right) ,$

$r = \left\{3 \cdot {4}^{\left(n + 1\right) + 1}\right\} \div \left\{3 \cdot {4}^{n + 1}\right\} .$

$\Rightarrow r = 4.$