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OlympiadBench / 2287 / Suppose that a and r are real numbers. A geometric sequence with first term…

Problem

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

answer type

Expression

is multiple answer

true

language

English

question

Suppose that

a

and

r

are real numbers. A geometric sequence with first term

a

and common ratio

r

has 4 terms. The sum of this geometric sequence is

6+6 \sqrt{2}

. A second geometric sequence has the same first term

a

and the same common ratio

r

, but has 8 terms. The sum of this second geometric sequence is

30+30 \sqrt{2}

. Determine all possible values for

a

. (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, called the common ratio. For example,

3,-6,12,-24

are the first four terms of a geometric sequence.)

Plain-text mathematical notation (without MathML)

Suppose that a and r are real numbers. A geometric sequence with first term a and common ratio r has 4 terms. The sum of this geometric sequence is 6+6√(2). A second geometric sequence has the same first term a and the same common ratio r, but has 8 terms. The sum of this second geometric sequence is 30+30√(2). Determine all possible values for a.

(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, called the common ratio. For example, 3,−6,12,−24 are the first four terms of a geometric sequence.)

Original LaTeX notation

Suppose that $a$ and $r$ are real numbers. A geometric sequence with first term $a$ and common ratio $r$ has 4 terms. The sum of this geometric sequence is $6+6 \sqrt{2}$. A second geometric sequence has the same first term $a$ and the same common ratio $r$, but has 8 terms. The sum of this second geometric sequence is $30+30 \sqrt{2}$. Determine all possible values for $a$.

(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, called the common ratio. For example, $3,-6,12,-24$ are the first four terms of a geometric sequence.)

question type

Open-ended

subject

Math

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Source and history

upstream_id / Original problem ID: 2287
task_id: 89fdc0d7-a071-58fc-9d98-16b2b5e81fc0
task_key: OE~5fTO~5fmaths~5fen~5fCOMP--train--2287
task_revision_id: 2
config / split: OE_TO_maths_en_COMP / train

Official source

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