Question

Find the Three Numbers in GPFind the three numbers in GP, whose sum is 52 and the sum of whose products in pairs is 624?I am not able to understand the meaning of "products in pairs".

Answer 1

Let the three numbers in G.P are a/r, a, ar.       --------------- (*)

Given sum = 52 and product = 624 i.e

Sum = a(1+r/r) = 52       ---------- (i)

Product = a^2(1/r + r + 1)  = 624   ------------ (ii)


On solving(i) and (ii), we get

a(1+r/r) = 52
a^2(1/r + r + 1) = 624

a = 12, r = 3

Substitute these values in (*), we get

12/3, 12*3, 12

= 4, 36, 12


Hope this helps!

Answer 2

Let the three numbers in the question be a/r a and ar.
a/c to the question,
a/r+ a + ar=52...(1)
and the sum of the product of pairs i.e.,
a²/r + a²r+ a² =624.....(2)
a²(1/r + r +1)=624...(3)
from 1 we have 1/r + r + 1=52/a 
putting in 3,
a*52=624
a=12
putting in 1 we get,
12/r +12r =52 -1 2
12((r²+1/)r)=40
(r²+1)/r=10/3
3r²+3=10r
3r²-10r+3 =0
3r²-9r -r +3 =0
3r(r-3) -1(r-3)=0
(3r-1)(r-3)=0
So, r=1/3 or 3.
putting values of a and r in 1,
we get,
the numbers to be,
4,12,36 or 36,12,4.
Hope this helps.