Question

Three numbers are in G.P. Their product is 64 and sum is 124/5. Find these numbers.

Answer 1

SOLUTION

GIVEN

• Three numbers are in G.P

• Their product is 64

• Their sum is 124/5.

TO DETERMINE

The numbers

EVALUATION

Since the given three numbers are in GP ( Geometric Progression)

Let the numbers are

$\displaystyle \sf{ \frac{a}{r} \:, \: a \: , \: ar}$

So by the first condition

$\displaystyle \sf{ \frac{a}{r} \: \times \: a \: \times \: ar = 64}$

$\implies \sf{ {a}^{3} = 64}$

$\implies \sf{a = 4}$

By the second condition

$\displaystyle \sf{ \frac{a}{r} + a + ar = \frac{124}{5} }$

$\implies \displaystyle \sf{ \frac{4}{r} + 4 + 4r = \frac{124}{5} }$

$\implies \displaystyle \sf{ \frac{4}{r} + 4r = \frac{124}{5} - 4}$

$\implies \displaystyle \sf{ \frac{4}{r} + 4r = \frac{104}{5} }$

$\implies \displaystyle \sf{ \frac{1}{r} + r = \frac{26}{5} }$

$\implies \displaystyle \sf{ 5 {r}^{2} - 26r + 5 = 0 }$

$\implies \displaystyle \sf{ 5 {r}^{2} - 25r - r + 5 = 0 }$

$\implies \displaystyle \sf{ 5r(r - 5) - 1(r - 5) = 0}$

$\implies \displaystyle \sf{ (r - 5) (5r - 1) = 0}$

$\implies \displaystyle \sf{ r = 5 \: \: or \: \: \frac{1}{5} }$

$\displaystyle \sf{ when \: \: r = 5 }$

The three numbers are

$\displaystyle \sf{ \frac{4}{5} \: ,\: 4 \: , \: 20}$

$\displaystyle \sf{ when \: \: r = \frac{1}{5} }$

The three numbers are

$\displaystyle \sf{ 20 \: ,\: 4 \: , \: \frac{4}{5} }$

━━━━━━━━━━━━━━━━

Learn more from Brainly :-

1. Insert 3 geometric means between 2 and 512.

https://brainly.in/question/23878688

2. If 8 GM's are inserted between 3 and 4

then the product of 8 GM's

https://brainly.in/question/29050188