Is there a pair of integers (a, b) such that

Daniell Phillips 2022-01-07 Answered
Is there a pair of integers \(\displaystyle{\left({a},{b}\right)}\) such that \(\displaystyle{a},{x}_{{1}},{y}_{{1}},{b}\) is part of an arithmetic sequences and \(\displaystyle{a},{x}_{{2}},{y}_{{2}},{b}\) is part of a geometric sequence with \(\displaystyle{x}_{{1}},{x}_{{2}},{y}_{{1}},{y}_{{2}}\) all integers?

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Jim Hunt
Answered 2022-01-08 Author has 1929 answers

It is given that \(\displaystyle{x}_{{1}},{x}_{{2}},{y}_{{1}},{y}_{{2}}\) are all integers.
\(\displaystyle{a},{x}_{{1}},{y}_{{1}},{b}\) is part of an arithmetic sequences.
Thus, \(\displaystyle{x}_{{1}}−{a}={y}_{{1}}−{x}_{{1}}\) the common difference is same in arithmetic sequences.
\(\displaystyle{x}_{{1}}−{a}={y}_{{1}}−{x}_{{1}}\)
\(\displaystyle{2}{x}_{{1}}={y}_{{1}}+{a}\)
\(\displaystyle{a}={2}{x}_{{1}}−{y}_{{1}}\)
So, \(\displaystyle{a}\) is integer as \(\displaystyle{x}_{{1}}\) and \(\displaystyle{y}_{{1}}\) are integers. (1)
\(\displaystyle{y}_{{1}}−{x}_{{1}}={b}−{y}_{{1}}\) the common difference is same in arithmetic sequences.
\(\displaystyle{y}_{{1}}−{x}_{{1}}={b}−{y}_{{1}}\)
\(\displaystyle{2}{y}_{{1}}={x}_{{1}}+{b}\)
\(\displaystyle{b}={2}{y}_{{1}}−{x}_{{1}}\)
So, \(\displaystyle{b}\) is integer as \(\displaystyle{x}_{{1}}\) and \(\displaystyle{y}_{{1}}\) are integers. (2)
It is given that \(\displaystyle{a},{x}_{{2}},{y}_{{2}},{b}\) is part of a geometric sequence. Hence, \(\displaystyle{\frac{{{x}_{{2}}}}{{{a}}}}={\frac{{{y}_{{2}}}}{{{x}_{{2}}}}}\) the common ratio is same in geometric sequence.
\(\displaystyle{x}_{{2}}{x}_{{2}}={a}{y}_{{2}}\)
\(\displaystyle{a}={\frac{{{{x}_{{2}}^{{{2}}}}}}{{{y}_{{2}}}}}\)
\(\displaystyle{x}_{{2}}=\sqrt{{{a}{y}_{{2}}}}\)
Similarly, \(y_2=\sqrt{bx_2}\)
Also, \(\displaystyle{\frac{{{x}_{{2}}}}{{{a}}}}={\frac{{{b}}}{{{y}_{{2}}}}}\) the common ratio is same in geometric sequence.
\(\displaystyle{a}{b}={x}_{{2}}{y}_{{2}}\)
\(\displaystyle{a}{b}\) is an integer as \(\displaystyle{x}_{{2}}\) and \(\displaystyle{y}_{{2}}\) are integerand the product of two integer is an integer.
Thus, from equations it is proved that there is a pair of integers \(\displaystyle{\left({a},{b}\right)}\).

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