What are the numbers that come next in these sequences: 3,3,6,9,15,24?

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Answer 1 · 2016-04-23T23:31:24

#39, 63, 102,...#

#a_n = 3F_n = (3(phi^n - (-phi)^(-n)))/sqrt(5)#

This is #3# times the standard Fibonacci sequence.

Each term is the sum of the two previous terms, but starting with #3, 3#, instead of #1, 1#.

The standard Fibonnaci sequence starts:

#1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987,...#

The terms of the Fibonacci sequence can be defined iteratively as:

#F_1 = 1#

#F_2 = 1#

#F_(n+2) = F_n + F_(n+1)#

The general term can also be expressed by a formula:

#F_n = (phi^n - (-phi)^(-n))/sqrt(5)#

where #phi = 1/2+sqrt(5)/2 ~~ 1.618033988#

So the formula for a term of our example sequence can be written:

#a_n = 3F_n = (3(phi^n - (-phi)^(-n)))/sqrt(5)#