In a $G . P$ if $T_{p - 1} + T_{p + 1} = 3 T_{p}$ then prove that the common ratio of $G . P$ is an irrational number.

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Solution

$\frac{T_{p - 1}}{T_{p}} + \frac{T_{p + 1}}{T_{p}} = 3 ∴ \frac{1}{r} + r = 3$
or $r^{2} - 3 r + 1 = 0$
$∴ r = \frac{3 \pm \sqrt{5}}{2}$ i.e. irrational.

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