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

Apr 23, 2016

$39 , 63 , 102 , \ldots$

${a}_{n} = 3 {F}_{n} = \frac{3 \left({\phi}^{n} - {\left(- \phi\right)}^{- n}\right)}{\sqrt{5}}$

Explanation:

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:

$1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 , 55 , 89 , 144 , 233 , 377 , 610 , 987 , \ldots$

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:

${F}_{n} = \frac{{\phi}^{n} - {\left(- \phi\right)}^{- n}}{\sqrt{5}}$

where $\phi = \frac{1}{2} + \frac{\sqrt{5}}{2} \approx 1.618033988$

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

${a}_{n} = 3 {F}_{n} = \frac{3 \left({\phi}^{n} - {\left(- \phi\right)}^{- n}\right)}{\sqrt{5}}$