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, 2010

Number theory Difficulty 3.0 AMC 10/12 Find the answer Canada

In a sequence of $10$ terms, the first term is $1$ , the second term is $x$ , and each term after the second is the sum of the previous two terms. For example, if $x = 11$ , the sequence would be $1, 11, 12, 23, 35, 58, 93, 151, 244, 395$ . For some values of $x$ , the number $463$ appears in the sequence. If $x$ is a positive integer, what is the sum of all the values of $x$ for which $463$ appears in the sequence?

(A) $1156$ (B) $1296$ (C) $1248$ (D) $1528$ (E) $1283$

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Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project. Solutions are the publisher's, linked not copied.