Three numbers are in AP, and their sum is 15. If 1, 4, 19 be added to them respectively, then they are in GP. Find the numbers.
To find: The numbers
Given: Three numbers are in A.P. Their sum is 15
Formula used: When a,b,c are in GP, b2 = ac
Let the numbers be a - d, a, a + d
According to first condition
a + d + a +a – d = 15
⇒ 3a = 15
⇒ a = 5
Hence numbers are 5 - d, 5, 5 + d
When 1, 4, 19 be added to them respectively then the numbers become –
5 – d + 1, 5 + 4, 5 + d + 19
⇒ 6 – d, 9, 24 + d
The above numbers are in GP
Therefore, 92 = (6 – d) (24 + d)
⇒ 81 = 144 – 24d +6d – d2
⇒ 81 = 144 – 18d – d2
⇒ d2 + 18d – 63 = 0
⇒ d2 + 21d – 3d – 63 = 0
⇒ d (d + 21) -3 (d + 21) = 0
⇒ (d – 3) (d + 21) = 0
⇒ d = 3, Or d = -21
Taking d = 3, the numbers are
5 - d, 5, 5 + d = 5 - 3, 5, 5 + 3
= 2, 5, 8
Taking d = -21, the numbers are
5 - d, 5, 5 + d = 5 – (-21), 5, 5 + (-21)
= 26, 5, -16
Ans) We have two sets of triplet as 2, 5, 8 and 26, 5, -16.