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Properties of Iota

If a,a2+2 a...

Question

If $a, a^{2} + 2$ and $a^{3} + 10$ are in $G . P$ , then find the value(s) of $a$ .

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Solution

$a, a^{2} + 2, a^{3} + 10$ are in G.P
then $\frac{a^{2} + 2}{a} = \frac{a^{3} + 10}{a^{2} + 2} a^{4} + 4 a^{2} + 4 = a^{4} + 10 a 4 a^{2} + 4 - 10 a = 0 4 a^{2} - 8 a - 2 a + 4 = 0 4 a (a - 2) - 2 (a - 2) = 0 (4 a - 2) (a - 2) = 0$
$a = 2$ or $a = 1 / 2$

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