1. [SOLVED] Sequences

What sequence goes in this pattern 1, 2, 5, 13, 34

2. Originally Posted by Liu997
What sequence goes in this pattern 1, 2, 5, 13, 34
Do you know of the Fibonnaci Sequence?

$\displaystyle F_{n + 2} = F_{n + 1} + F_n$, with $\displaystyle F_0 = 0$ and $\displaystyle F_1 = 1$.

In other words, each term is found by adding the previous two.

$\displaystyle 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, \dots$.

Can you see that your sequence is the Fibonnaci sequence with each even subscript removed (assuming you start at $\displaystyle F_0$?

3. Hello, Liu997!

What sequence goes in this pattern? .$\displaystyle 1,\: 2,\: 5,\: 13,\: 34,\:\hdots$
I too recognized the alternate terms of the Fibonacci sequence,
. . but what if I didn't?

The expected answer is probably: .$\displaystyle a_{n+1}\;=\;3a_n - a_{n-1}$

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The "other" terms of the Fibonacci sequence has the same recurrence:

. . $\displaystyle 1,\:3,\:8,\:21,\:55,\:144,\:\hdots \qquad a_{n+1} \:=\:3a_n - a_{n-1}$

Why am I surprised?
.