Question

The sum of three numbers in AP is 18. If the product of first and third number is five times the common difference, find the numbers.

HINT: Let these numbers be (a – d), a and (a + d).

Answer 1

Given:

Three numbers are in AP and their sum is 18.

The product of the first and third equals five times the common difference.

Step-wise calculation:

1. Let the three numbers be (a – d), a, (a + d).

2. Sum condition:

(a – d) + a + (a + d) = 3a = 18

⇒ a = 6

3. Product condition:

(a – d)(a + d) = a² – d²

= 5d

Substitute a = 6:

36 – d² = 5d

4. Rearrange to a standard quadratic:

d² + 5d – 36 = 0

5. Solve:

Discriminant Δ = 5² – 4(1)(–36) 

= 25 + 144

= 169

⇒ `sqrt(Δ) = 13`

So, `d = (-5 ± 13)/2`

⇒ `d = (8)/2 = 4` or `d = (-18)/2 = -9`

6. For d = 4:

Numbers = (6 – 4, 6, 6 + 4)

= (2, 6, 10)

For d = –9:

Numbers = (6 – (–9), 6, 6 + (–9))

= (15, 6, –3)

The three numbers are either (2, 6, 10) or (15, 6, –3).