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3;3;6;9;15;... are the first five terms of a quadratic pattern. Write down the value of the sixth term (T6) of the pattern. | 447 views

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The sixth term is 24 as shown below.

\begin{aligned} 3,3,6,9,15 & \\ 3+3 &=6 \\ 3+6 &=9 \\ 6+9 &=15 \\ 9+15 &=24 \\ \therefore 3 ; 3 ; 6 ; 9 ; 15 ; 24 \end{aligned} by Diamond (58.4k points)
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$a_{n}=3 F_{n}=\frac{3\left(\phi^{n}-(-\phi)^{-n}\right)}{\sqrt{5}}$
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:
\begin{aligned} &F_{1}=1 \\ &F_{2}=1 \\ &F_{n+2}=F_{n}+F_{n+1} \end{aligned}
The general term can also be expressed by a formula:
$F_{n}=\frac{\phi^{n}-(-\phi)^{-n}}{\sqrt{5}}$

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:
$a_{n}=3 F_{n}=\frac{3\left(\phi^{n}-(-\phi)^{-n}\right)}{\sqrt{5}}$
by Platinum (101k points)

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