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OlympiadBench / 2300 / The numbers a₁,a₂,a₃,… form an arithmetic sequence with…

Problem

Answer published by the source. Consult the official source to check your work against its answer.

answer type

Numerical

is multiple answer

false

language

English

question

The numbers

a_{1}, a_{2}, a_{3}, \ldots

form an arithmetic sequence with

a_{1} \neq a_{2}

. The three numbers

a_{1}, a_{2}, a_{6}

form a geometric sequence in that order. Determine all possible positive integers

k

for which the three numbers

a_{1}, a_{4}, a_{k}

also form a geometric sequence in that order. (An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding a constant. For example, 3, 5, 7, 9 are the first four terms of an arithmetic sequence. A geometric sequence is a sequence in which each term after the first is obtained from the previous term by multiplying it by a non-zero constant. For example,

3,6,12

is a geometric sequence with three terms.)

Plain-text mathematical notation (without MathML)

The numbers a₁,a₂,a₃,… form an arithmetic sequence with a₁≠a₂. The three numbers a₁,a₂,a₆ form a geometric sequence in that order. Determine all possible positive integers k for which the three numbers a₁,a₄,a_(k) also form a geometric sequence in that order.

(An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding a constant. For example, 3, 5, 7, 9 are the first four terms of an arithmetic sequence.

A geometric sequence is a sequence in which each term after the first is obtained from the previous term by multiplying it by a non-zero constant. For example, 3,6,12 is a geometric sequence with three terms.)

Original LaTeX notation

The numbers $a_{1}, a_{2}, a_{3}, \ldots$ form an arithmetic sequence with $a_{1} \neq a_{2}$. The three numbers $a_{1}, a_{2}, a_{6}$ form a geometric sequence in that order. Determine all possible positive integers $k$ for which the three numbers $a_{1}, a_{4}, a_{k}$ also form a geometric sequence in that order.

(An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding a constant. For example, 3, 5, 7, 9 are the first four terms of an arithmetic sequence.

A geometric sequence is a sequence in which each term after the first is obtained from the previous term by multiplying it by a non-zero constant. For example, $3,6,12$ is a geometric sequence with three terms.)

question type

Open-ended

subject

Math

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Source and history

upstream_id / Original problem ID: 2300
task_id: 654425b2-2c8d-56ab-a79a-d959c29d9737
task_key: OE~5fTO~5fmaths~5fen~5fCOMP--train--2300
task_revision_id: 1
config / split: OE_TO_maths_en_COMP / train

Official source

initial import