# How do you find the explicit formula for a geometric sequence 9,19, 39,79,159,...?

Mar 16, 2016

${b}_{n} = 10 \cdot {2}^{n} - 1 , n = 0 , 1 , 2 , \ldots$

#### Explanation:

A geometric sequence is a sequence in which there is a common ratio between successive terms. The given sequence is not a geometric sequence, as, for example, $\frac{19}{9} \ne \frac{39}{19}$.

The sequence is close to a geometric sequence, however. If we add $1$ to each term, then the sequence becomes $10 , 20 , 40 , 80 , 160 , \ldots$ which is a geometric sequence with initial term $10$ and common ratio $2$.

The general term for a geometric series with common ratio $r$ and initial term ${a}_{0}$ is

${a}_{n} = {a}_{0} {r}^{n} , n = 0 , 1 , 2 , \ldots$

In that case, if the general term for the given series is ${b}_{n}$, we have

${b}_{n} + 1 = 10 \cdot {2}^{n}$

Subtracting $1$ gives us our result:

${b}_{n} = 10 \cdot {2}^{n} - 1 , n = 0 , 1 , 2 , \ldots$