find-three-nos-in-AP-whose-product-is-equal-to-the-square-of-their-sum-

Question Number 31281 by MURALI last updated on 05/Mar/18

f i n d t h r e e n o s i n A P w h o s e p r o d u c t

i s e q u a l t o t h e s q u a r e o f t h e i r s u m .

Answered by Rasheed.Sindhi last updated on 07/Mar/18

Let the middle number is a

and the common difference is d

∴ Numbers are a - d, a, a + d

(a - d) (a) (a + d) = {(a - d) + (a) + (a + d)}^{2}

a (a^{2} - d^{2}) = {(3 a)}^{2}

a^{3} - {9 a}^{2} - {ad}^{2} = 0

a (a^{2} - 9 a - d^{2}) = 0

^{∙ 1} a = 0 ∣^{∙ 2} a^{2} - 9 a - d^{2} = 0

^{∙ 1} As a = 0 is free of d so d may be any number .

Hence in general - d, 0, d are required nos .

For example - 3, 0, 3; - 4, 0, 4 etc .

^{∙ 2} a^{2} - 9 a - d^{2} = 0

a = \frac{9 \pm \sqrt{81 + {4 d}^{2}}}{2}

In general

\frac{9 \pm \sqrt{81 + {4 d}^{2}}}{2} - d, a, \frac{9 \pm \sqrt{81 + {4 d}^{2}}}{2} + d

are required numbers for any number d .

If a, d \in Z, d = \pm 6, \pm 20 work .

For d = 6 required numbers are :

6, 12, 18 or - 9, - 3, 3

For d = 20 required numbers are :

5, 25, 45 or - 36, - 16, 4