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MATH / A sequence of positive integers with a₁=1 and a₉+a₁₀=646 is formed so that the first three terms are in geometric progression, the second, third, and fourth terms are in arithmetic…

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problem

A sequence of positive integers with a1=1a_1 = 1 and a9+a10=646a_9+a_{10}=646 is formed so that the first three terms are in geometric progression, the second, third, and fourth terms are in arithmetic progression, and, in general, for all n≥1n\ge1, the terms a2n−1a_{2n-1}, a2na_{2n}, and a2n+1a_{2n+1} are in geometric progression, and the terms a2na_{2n}, a2n+1a_{2n+1}, and a2n+2a_{2n+2} are in arithmetic progression. Let ana_n be the greatest term in this sequence that is less than 1000. Find n+ann+a_n.
Plain-text mathematical notation (without MathML)
A sequence of positive integers with a₁=1 and a₉+a₁₀=646 is formed so that the first three terms are in geometric progression, the second, third, and fourth terms are in arithmetic progression, and, in general, for all n≥1, the terms a_(2n−1), a_(2n), and a_(2n+1) are in geometric progression, and the terms a_(2n), a_(2n+1), and a_(2n+2) are in arithmetic progression. Let a_(n) be the greatest term in this sequence that is less than 1000.  Find n+a_(n).
Original LaTeX notation
A sequence of positive integers with $a_1 = 1$ and $a_9+a_{10}=646$ is formed so that the first three terms are in geometric progression, the second, third, and fourth terms are in arithmetic progression, and, in general, for all $n\ge1$, the terms $a_{2n-1}$, $a_{2n}$, and $a_{2n+1}$ are in geometric progression, and the terms $a_{2n}$, $a_{2n+1}$, and $a_{2n+2}$ are in arithmetic progression. Let $a_n$ be the greatest term in this sequence that is less than 1000.  Find $n+a_n$.

level

Level 3

type

Intermediate Algebra

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Source and history

task_id
a23e9c6a-bee5-5384-b81f-466ce32441e4
task_key
intermediate~5falgebra--test--a23e9c6a-bee5-5384-b81f-466ce32441e4
task_revision_id
1
config / split
intermediate_algebra / test

Official source

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