Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11

# The Product of Three Numbers in G.P. is 125 and the Sum of Their Products Taken in Pairs is 87 1 2 . Find Them. - Mathematics

The product of three numbers in G.P. is 125 and the sum of their products taken in pairs is $87\frac{1}{2}$ . Find them.

#### Solution

Let the required numbers be $\frac{a}{r},\text { a and ar } .$ Product of the G.P. = 125

$\Rightarrow a^3 = 125$

$\Rightarrow a = 5$

Sum of the products in pairs = $87\frac{1}{2} = \frac{175}{2}$

$\Rightarrow \frac{a}{r} \times a + a \times ar + ar \times \frac{a}{r} = \frac{175}{2}$

$\Rightarrow \frac{a^2}{r} + a^2 r + a^2 = \frac{175}{2}$

$\text {Substituting the value of a }$

$\Rightarrow \frac{25}{r} + 25r + 25 = \frac{175}{2}$

$\Rightarrow 50 r^2 + 50r + 50 = 175r$

$\Rightarrow 50 r^2 - 125r + 50 = 0$

$\Rightarrow 25(2 r^2 - 5r + 2) = 0$

$\Rightarrow 2 r^2 - 4r - r + 2 = 0$

$\Rightarrow 2r(r - 2) - 1(r - 2) = 0$

$\Rightarrow (2r - 1)(r - 2) = 0$

$\therefore r = \frac{1}{2}, 2$

Hence, the G.P. for a = 5 and r = $\frac{1}{2}$ is 10, 5 and $\frac{5}{2}$ .

And, the G.P. for a = 5 and r 2 is $\frac{5}{2}$ , 5 and 10.

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 20 Geometric Progression
Exercise 20.2 | Q 4 | Page 16