2024 AMC 12B Problem 18

Problem 18 of 25IntermediateAlgebra

The Fibonacci numbers are defined by

F_1 = 1,

F_2 = 1,

and

F_n = F_{n-1} + F_{n-2}

for

n \ge 3.

What is

\frac{F_2}{F_1} + \frac{F_4}{F_2} + \frac{F_6}{F_3} + \cdots + \frac{F_{20}}{F_{10}}?

Show solution

Solution

Since

F_{2k} = F_k L_k

where

L_k

is the

k

th Lucas number, each term

\dfrac{F_{2k}}{F_k} = L_k.

The sum is

\begin{gathered} L_1 + L_2 + \cdots + L_{10} \\ = 1 + 3 + 4 + 7 + 11 + 18 \\ {}+ 29 + 47 + 76 + 123 \\ = 319. \end{gathered}

(Equivalently,

L_1 + \cdots + L_{10}

= L_{12} - 3

= 322 - 3 = 319.

) Thus, the correct answer is B.