Difference between revisions of "2024 AMC 10B Problems/Problem 23"
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==Problem== | ==Problem== | ||
The Fibonacci numbers are defined by <math>F_1 = 1, F_2 = 1,</math> and <math>F_n = F_{n-1} + F_{n-2}</math> for <math>n \geq 3.</math> What is <cmath>{\frac{F_2}{F_1}} + {\frac{F_4}{F_2}} + {\frac{F_6}{F_3}} + ... + {\frac{F_{20}}{F_{10}}}?</cmath> | The Fibonacci numbers are defined by <math>F_1 = 1, F_2 = 1,</math> and <math>F_n = F_{n-1} + F_{n-2}</math> for <math>n \geq 3.</math> What is <cmath>{\frac{F_2}{F_1}} + {\frac{F_4}{F_2}} + {\frac{F_6}{F_3}} + ... + {\frac{F_{20}}{F_{10}}}?</cmath> |
Revision as of 05:39, 14 November 2024
Possible duplicate [1]
Contents
Problem
The Fibonacci numbers are defined by and
for
What is
Solution 1
Brute forcing gets you B) 319
Solution 2
Plug in a few numbers to see if there is a pattern. List out a few Fibonacci numbers, and then try them on the equation. You'll find that and
The pattern is that then ten fractions are in their own Fibonacci sequence with the starting two terms being
and
, which can be written as
for
The problem is asking for the sum of the ten terms
, and you arrive at the solution
~Cattycute
See also
2024 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 22 |
Followed by Problem 24 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions.