Question

Use the value of phi = 1.618 to predict the 23rd number in the Fibonacci sequence. The 22nd number in the sequence is 17,711.

- 10,946

- 17,711

- 22,897

- 28,656

Answer 1

Given:

$\phi=1.618$

22nd number in Fibonacci sequence = 17,711

To find:

The 23rd number in the Fibonacci sequence.

Solution:

The nth term of a Fibonacci sequence is:

$f_n=\dfrac{\phi^n-(1-\phi)^n}{\sqrt{5}}$

Substituting $\phi=1.618, n=21$, we get

$f_{21}=\dfrac{(1.618)^{21}-(1-1.618)^{21}}{\sqrt{5}}$

$f_{21}=\dfrac{(1.618)^{21}-(1-1.618)^{21}}{\sqrt{5}}$

$f_{21}=10941.1724024$

$f_{21}\approx 10941$

Now,

$f_{23}=f_{21}+f_{22}$

$f_{23}=10941+17711$

$f_{23}=28652$

It is about 28,656. Therefore, the correct option is D.