Question

sum of three numbers in AP is 6 and their square is 14 find the number​

Answer 1

Let the 3 numbers in A.P be

• (a - d)
• a
• (a + d)

Sum of 3 numbers = 6

➥ (a - d) + a + (a + d) = 6

↠ a - d + a + a + d = 6

↦ 3a = 6

⇢ a = 6 ÷ 3

∴ a = 2

Sum of their squares = 14

➥ (a - d)² + a² + (a + d)² = 14

⇒ (2 - d)² + 2² + (2 + d)² = 14

⇒ (4 + d² - 4d) + 4 + (4 + d² + 4d) = 14

⇒ 4 + d² - 4d + 4 + 4 + d² + 4d = 14

➝ 2d² + 12 = 14

➝ 2d² = 14 - 12

➝ 2d² = 2

➝ d² = 1

➝ d = √1

∴ d = ±1

Case 1:

If d = +1,

• (a - d) = 2 - 1 = 1
• a = 2
• (a + d) = 2 + 1 = 3

So, the numbers would be 1, 2, 3...

Case 2:

If d = -1,

• (a - d) = 2 - (-1) = 2 + 1 = 3
• a = 2
• (a + d) = 2 - 1 = 1

So, the numbers are 3, 2, 1...