# Thread: Formula for nth term of sequence

1. ## Formula for nth term of sequence

Suppose that $a_{1} = 3$ and $a_{n+1}=a_{n}\left(a_{n}+2\right)$. Find a formula for the nth term of this sequence.

2. ## Hint

Here's a hint: consider the fact that $(2^k-1)[(2^k-1)+2]=(2^k-1)(2^k+1)=2^{2k}-1$.

--Kevin C.

3. $a_{n+1}=a_{n}\left(a_{n}+2\right)\text{ implies }a_{n+1}+1=a_{n}\left(a_{n}+2\right)+1,\text{ that is }a_{n+1}+1=\left(a_{n}+1\right)^2$
Construct a new sequence by letting $b_n=a_n+1$, you can take it from here.

4. Hello, Pn0yS0ld13r!

I found the answer by inspection . . .

Suppose that $a_1 = 3$ and $a_{n+1}\:=\:a_n\left(a_n+2\right)$.

Find a formula for the $n^{th}$ term of this sequence.
I cranked out the first few terms . . .

. . $\begin{array}{|c|ccc|}
n & & a_n & \\ \hline
1 & 3 &=& 2^2-1 \\
2 & 15 &=& 2^4-1\\
3 & 255 &=& 2^8-1\\
4 & 65,\!535 &=& 2^{16}-1 \\
5 & 4,\!294,\!967,\!295 &=& 2^{32}-1 \\
\vdots & \vdots & & \vdots
\end{array}$

And saw that: . $a_n \;=\;2^{2^n}-1$