what is the 14th term in the geometric sequence 7, 14, 28, 56
You asked:
Determine the \(14^{\text{th}}\) term in the sequence \(7, 14, 28, 56, \ldots\).
MathBot Answer:
$$7,14,28,56,...$$The 14th term in this geometric sequence is $57344$.
The nth term in this sequence is given by the formula:
Explicit Formula: $$a_n=7 \cdot 2^{n - 1}$$
Recursive Formula: $$a_n=2 a_{n-1}, \text{where } a_{1}=7$$
$$\begin{matrix}7&&14&&28&&56\\[6pt]&\times2&&\times2&&\times2\\[6pt]\end{matrix}$$
Explicit Formula
The formula for a geometric sequence where $a_1$ is the 1st term, $r$ is the common ratio, and $n$ is the term number is $$a_n=a_1 r^{n-1}$$
Find $a_1$ and $r$: $$\begin{aligned} a_1&=7 \\ r&=2 \end{aligned}$$
The nth term rule is:$$\begin{aligned} a_n&=a_1 r^{n-1} \\ &=7 \left(2\right)^{n-1} \\ &=7 \cdot 2^{n - 1} \end{aligned}$$
Recursive Formula
The formula for a geometric sequence where $a_{n-1}$ is the (n-1)th term, $r$ is the common ratio, and $n>1$ is $$a_n=r a_{n-1}$$
Find $r$: $$\begin{aligned} r=2 \end{aligned}$$
The nth term rule is:$$\begin{aligned} a_n&=r a_{n-1} \\ &=\left(2\right) a_{n-1} \\ &=2 a_{n-1} \end{aligned}$$