The-sum-of-the-squares-of-three-distinct-real-numbers-which-are-in-GP-is-S-2-If-their-sum-is-S-then-

Question Number 7525 by Little last updated on 02/Sep/16

The sum of the squares of three distinct

real numbers which are in GP is S^{2} . If

their sum is α S, then

Commented by Rasheed Soomro last updated on 04/Sep/16

A s s u m e d t h a t α i s r e q u i r e d .

L e t a i s t h e i n i t i a l r e a l n u m b e r a n d r i s

c o m m o n r a t i o .

S o t h r e e d i s t i n c t r e a l n u m b e r s a r e

a, a r, {a r}^{2}

S u m o f s q u a r e s : a^{2} + a^{2} r^{2} + a^{2} r^{4} = S^{2} \dots \dots \dots . . (i)

S u m o f r e a l n u m b e r s : a + a r + {a r}^{2} = α S

O r S = \frac{a (1 + r + r^{2})}{α} \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots (i i)

F r o m (i) & (i i)

a^{2} (1 + r^{2} + r^{4}) = {(\frac{a (1 + r + r^{2})}{α})}^{2}

1 + r^{2} + r^{4} = \frac{{(1 + r + r^{2})}^{2}}{α^{2}}

α^{2} = \frac{{(1 + r + r^{2})}^{2}}{1 + r^{2} + r^{4}}