Question

Find three numbers in gp whose sum is 52 abd sum of whose products pair is 624

Answer 1

$\textbf{\underline{\underline{According\:to\:the\:Question}}}$

Assumption

Required Numbers

a , ar , ar²

Now,

(a + ar + ar²) = 52

${\boxed{\bigstar{{Taking\;Common}}}}$

a (1 + r + r²) = 52 ..... (1)

Also,

a × ar + ar × ar² + a × ar² = 624

${\boxed{\bigstar{{Taking\;Common}}}}$

a²r(1 + r + r²) = 624 ..... (2)

Dividing (2) by (1)

ar = 12

${\boxed{\sf\:{a=\dfrac{12}{r}}}}$

Putting value in (1) we get,

${\boxed{\sf\:{\dfrac{12}{r}\times(1+r+r^2)=52}}}$

= 3(1 + r + r²) = 13r

= 3r² - 10r + 3 =0

= (3r - 1)(r - 3) = 0

${\boxed{\sf\:{r=\dfrac{1}{3}\;or\;r=3}}}$

Here we get,

a = 36 or a = 4

Hence,

(36 , 12 , 4) or (4 , 12 , 36)