Question

The sum of three numbers in A.P. is 24 and sum of their squares is 194 . Find the numbers?

Answer 1

Answer:

7 , 8 & 9

Step-by-step explanation:

Given :

The sum of three numbers in A.P is 24,.

The sum of their squares is 194.

To find :

The 3 numbers.

Solution :

As they are in A.P , they must be of the form : (a - d) , a & (a + d)

Hence, their sum :

⇒ (a - d) + a + (a - d) = 24

⇒ 3a = 24 ⇒ a = 8,.

Hence, the middle (or 2nd) term is 8,.

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The sum of their squares will be,

⇒ (a - d)² + a² + (a + d)² = 194

⇒ (8 - d)² + 8² + (8 + d)² = 194

⇒ (8 - d)² + (8 + d)² = 194 - 64

⇒ 2(8² + d²) = 130

⇒ 128 + 2d² = 130

⇒ 2d² = 13 0 - 128 = 2

⇒ 2d² = 2

⇒ d² = 1 ⇒ d = ±1

Hence, the numbers must be ,

(8 ∓ 1) , 8 , (8 ± 1)

⇒ The numbers are 7 , 8 & 9 respectively,.

Answer 2

$\huge \bold{hey \: there}$

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

According to the question,

➡a-d+a+a+d=24

➡3a=24

➡a=8

Now,

Sum of their squares=194

➡(a-d)²+(a)²+(a+d)²=194

➡(8-d)²+(8)²+(8+d)²=194

➡64+d²-16d+64+64+d²+16d=194

➡192+2d²=194

➡2d²=2

➡d²=1

➡d=±1

Thus, the the numbers are

➡a-d=8-1=7

➡a=8

➡a+d=9

↪7,8,9....