MCQ
If a be A.M. and p, q be two G.M.'s between two numbers, then 2A is equal to:
- A$\frac{\text{p}^3+\text{q}^3}{\text{pq}}$
- B$\frac{\text{p}^3-\text{q}^3}{\text{pq}}$
- C$\frac{\text{p}^2+\text{q}^2}{2}$
- D$\frac{\text{pq}}{2}.$
Solution:
Let the two positive numbers be a and b.
a, A and b are in A.P.
$\therefore2\text{A}=\text{a}+\text{b}\cdots(\text{i})$
Also, a, p, q and b are in G.P.
$\therefore\text{r}=\Big(\frac{\text{b}}{\text{a}}\Big)^{\frac{1}{3}}$
Again, p = ar and q = ar2 ____(ii)
Now, 2A = a + b [From (i)]
$=\text{a}+\text{a}\Big(\frac{\text{b}}{\text{a}}\Big)$
$=\text{a}+\text{a}\Bigg(\Big(\frac{\text{b}}{\text{a}}\Big)^{\frac{1}{3}}\Bigg)^3$
$=\text{a}+\text{ar}^3$
$=\frac{(\text{ar})^2}{\text{ar}^2}+\frac{\big(\text{ar}^2\big)^2}{\text{ar}}$
$=\frac{\text{p}^2}{\text{q}}+\frac{\text{q}^2}{\text{p}}$ [Using (ii)]
$=\frac{\text{p}^3+\text{q}^3}{\text{pq}}$
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