Question

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.

Answer 1

The sixth term is 24 as shown below.

\(\begin{aligned} 3,3,6,9,15 & \\ 3+3 &=6 \\ 3+6 &=9 \\ 6+9 &=15 \\ 9+15 &=24 \\ \therefore 3 ; 3 ; 6 ; 9 ; 15 ; 24 \end{aligned}\)

Answer 2

\[
a_{n}=3 F_{n}=\frac{3\left(\phi^{n}-(-\phi)^{-n}\right)}{\sqrt{5}}
\]
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:
\[
a_{n}=3 F_{n}=\frac{3\left(\phi^{n}-(-\phi)^{-n}\right)}{\sqrt{5}}
\]