# How do you write the first five terms of the geometric sequence a_1=7, a_(k+1)=2a_k and determine the common ratio and write the nth term of the sequence as a function of n?

May 13, 2017

Common Ratio: $2$
Explicit formula: ${a}_{n} = 7 \left({2}^{n - 1}\right)$

#### Explanation:

Given the first term, ${a}_{1} = 7$ and the recursive formula for the geometric sequence, ${a}_{k + 1} = 2 {a}_{k}$, we know the common ratio must be $2$, since the recursive formula multiplies $2$ to the ${k}^{t h}$ term to get the ${\left(k + 1\right)}^{t h}$ term.

Since we know the common ratio is $2$ and the first term is $7$, we can write our explicit formula for the geometric sequence using the general form:
${a}_{n} = {a}_{1} \left({r}^{n - 1}\right)$

By substituting our known values, we get:
${a}_{n} = 7 \left({2}^{n - 1}\right)$ which allows us to find the nth term as a function of n