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प्रश्न
If A be one A.M. and p, q be two G.M.'s between two numbers, then 2 A is equal to
पर्याय
(a) \[\frac{p ^3 + q^3}{pq}\]
(b) \[\frac{p^3 - q^3}{pq}\]
(c) \[\frac{p^2 + q^2}{2}\]
(d) \[\frac{pq}{2}\]
उत्तर
(a) \[\frac{p^3 + q^3}{pq}\]
\[\text{ Let the two positive numbers be a and b } . \]
\[ \text{ a, A and b are in A . P }. \]
\[ \therefore 2A = a + b (i)\]
\[\text{ Also, a, p, q and b are in G . P } . \]
\[ \therefore r = \left( \frac{b}{a} \right)^\frac{1}{3} \]
\[\text{ Again, p = ar and } q = a r^2 . (ii)\]
\[\text{ Now }, 2A = a + b \left[ \text{ From } (i) \right]\]
\[ = a + a\left( \frac{b}{a} \right)\]
\[ = a + a \left( \left( \frac{b}{a} \right)^\frac{1}{3} \right)^3 \]
\[ = a + a r^3 \]
\[ = \frac{\left( ar \right)^2}{a r^2} + \frac{\left( a r^2 \right)^2}{ar}\]
\[ = \frac{p^2}{q} + \frac{q^2}{p} \left[ \text{ Using } (ii) \right]\]
\[ = \frac{p^3 + q^3}{pq}\]
\[\]
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