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

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

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Answer 1 · 2016-03-16T23:36:58

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

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:

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

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

1 Answer

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