Difference between revisions of "2007 iTest Problems/Problem 53"
Rockmanex3 (talk | contribs) (Solution to Problem 53) |
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+ | ''The following problem is from the Ultimate Question of the [[2007 iTest]], where solving this problem required the answer of a previous problem. When the problem is rewritten, the T-value is substituted.'' | ||
+ | |||
== Problem == | == Problem == | ||
− | + | Three distinct positive Fibonacci numbers, all greater than <math>1536</math>, are in arithmetic progression. Let <math>N</math> be the smallest possible value of their sum. Find the remainder when <math>N</math> is divided by <math>2007</math>. | |
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+ | ==Solution== | ||
+ | By definition, for a Fibonacci number, <math>a_1 = a_2 = 1</math> and <math>a_n = a_{n-1} + a_{n-2}</math>. From the definition, <math>a_{n+1} = a_n + a_{n-1}</math>. That means the numbers <math>a_{n-2}</math>, <math>a_n</math>, and <math>a_{n+1}</math> are in arithmetic progression with common difference <math>a_{n-1}</math>. | ||
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+ | Writing out the Fibonacci numbers, the first numbers that come after <math>1536</math> are <math>1597</math>, <math>2584</math>, <math>4181</math>, and <math>6765</math>. That means the desired three numbers are <math>1597</math>, <math>4181</math>, and <math>6765</math>. The sum of the three numbers is <math>12543</math>, and the remainder after dividing by <math>2007</math> is <math>\boxed{501}</math>. | ||
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+ | ==See Also== | ||
+ | {{iTest box|year=2007|num-b=52|num-a=54}} |
Latest revision as of 01:05, 25 June 2018
The following problem is from the Ultimate Question of the 2007 iTest, where solving this problem required the answer of a previous problem. When the problem is rewritten, the T-value is substituted.
Problem
Three distinct positive Fibonacci numbers, all greater than , are in arithmetic progression. Let be the smallest possible value of their sum. Find the remainder when is divided by .
Solution
By definition, for a Fibonacci number, and . From the definition, . That means the numbers , , and are in arithmetic progression with common difference .
Writing out the Fibonacci numbers, the first numbers that come after are , , , and . That means the desired three numbers are , , and . The sum of the three numbers is , and the remainder after dividing by is .
See Also
2007 iTest (Problems) | ||
Preceded by: Problem 52 |
Followed by: Problem 54 | |
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