Pattern Detective! Part 2

Relevant wiki: Fibonacci Sequence

1 + 4 = 5

5 + 4 = 9

9 + 5 = 14

14 + 9 = 23

23 + 14 = 37

Next number is 37

In each case, the next number is the the number added to its predecessor. So, the answer is $14 + 23 = \color{#69047E}{\boxed{37}}$ .

As you can see from the sequence, $a_{n}=a_{n-1}+a_{n-2}$ and $a_{1}=1, a_{2}=4$ . Therefore, the next term would be $14+23=\color{#69047E}{37}$ .

A closed form would be $a_{n}=\frac{3L_{n}-F_{n}}{2}$ , where $L_{n}$ denotes the nth Lucas number and $F_{n}$ denotes the nth Fibonacci number .