Question

The sum of 3 nos in ap is 24 and sum of their squares is 194 find nos

Answer 1

let the terms of in ap be x-a ,x , and x+a

x-a+x+x+a=24
3x=24
x=8
(x-a)2+x2+(x+a)2=194
x2+a2-2ax+x2+x2+a2+2ax=194
3x2+2a2=194
3*8*8+2a2=194
192+2a2=194
2a2=2
a2=1
a=1
numbers are 8-1,8,8+1
7 , 8 ,9

Answer 2

Let the three numbers be a - d, a, a + d.

a - d, a, a + d are in AP with common difference d.

Given

Sum of numbers is 24

⇒(a - d) + a + ( a + d) = 24

⇒3a = 24

⇒ a = 8.

Sum of squares of the numbers is 194

⇒(a - d)² + a²+ ( a + d)² = 194

We have found that, a = 8

⇒(8 - d)² + 8²+ ( 8 + d)² = 194

⇒64 + d² - 16d + 64 + 64 + d² + 16d = 194

⇒(64 + 64 + 64) + 2d² + ( 16 d - 16d) = 194

⇒192 + 2d² = 194

⇒ 2d² = 194 - 192

⇒ 2d² = 2

⇒d² = 1

Either we take d = 1, - 1 doesn't matter as we have a - d and a + d.

Therefore, The numbers are 7, 8, 9.