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

1 Answer
sente
Mar 16, 2016

#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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