Question

In a A.P., the sum of the three consecutive terms is 24 and the sum of their squares is 194. Find the numbers.

Answer 1

Step 1: Using the sum of terms

(a − d) + a + (a + d) = 24

3a = 24 

a = 8

Step 2: Using the sum of squares

(a − d)² + a² + (a + d)² = 194

(8 − d)² + 8² + (8 + d)² = 194

(64 − 16d + d²) + 64 + (64 + 16d + d²) = 194

64 + 64 + 64 − 16d + 16d + d² + d² = 194

192 + 2d² = 194

2d² = 194 − 192

2d² = 2

d² = 1

d = ±1

Step 3: Find the terms

If d = 1: The terms are: 8 − 1, 8, 8 + 1 ⟹ 7, 8, 9

If d = −1: The terms are: 8 − (−1), 8, 8 + (−1) ⟹ 9, 8, 7

The three terms of the A.P. are: 7, 8, 9 or 9, 8, 7