Difference between revisions of "2024 AMC 10B Problems/Problem 24"
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==Problem 18== | ==Problem 18== | ||
The Fibonacci numbers are defined by <math>F_1=1,</math> <math>F_2=1,</math> and <math>F_n=F_{n-1}+F_{n-2}</math> for <math>n\geq 3.</math> What is<cmath>\dfrac{F_2}{F_1}+\dfrac{F_4}{F_2}+\dfrac{F_6}{F_3}+\cdots+\dfrac{F_{20}}{F_{10}}?</cmath> | The Fibonacci numbers are defined by <math>F_1=1,</math> <math>F_2=1,</math> and <math>F_n=F_{n-1}+F_{n-2}</math> for <math>n\geq 3.</math> What is<cmath>\dfrac{F_2}{F_1}+\dfrac{F_4}{F_2}+\dfrac{F_6}{F_3}+\cdots+\dfrac{F_{20}}{F_{10}}?</cmath> |
Revision as of 05:49, 14 November 2024
Contents
[hide]Problem 18
The Fibonacci numbers are defined by and for What is
Solution 1
The first terms
so the answer is .
Solution 2
Define new sequence
A= and B =
Per characteristic equation, itself is also Fibonacci type sequence with starting item
then we can calculate the first 10 items using
so the answer is .
See also
2024 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
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 |
2024 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 17 |
Followed by Problem 19 |
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 12 Problems and Solutions |
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