Question

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

Answer 1

`b_n = 10*2^n-1, n = 0, 1, 2, ...`

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, `19/9 != 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, ...` 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_0r^n, n = 0, 1, 2, ...`

In that case, if the general term for the given series is `b_n`, we have

`b_n + 1 = 10*2^n`

Subtracting `1` gives us our result:

`b_n = 10*2^n-1, n = 0, 1, 2, ...`