Question

5) the sum of three decreasing numbers in a. p. is 27. if -1,-1,3 are added to them respectively, the resulting series is a G.P. find the numbers. solve the question.

Answer 1

Step-by-step explanation:

Given:

• There are 3 decreasing numbers in an AP => common difference d < 0 ( negative )
• Sum of those numbers = 27
• If -1, -1 & 3 are added to the numbers respectively, resulting series will be GP

Solution:

Let the numbers be a, b & c respectively

a, b & c are in AP

=> 2b = a + c

Sum of a, b & c is 27

=> a + b + c = 27

=> 2b + b = 27

=> 3b = 27

=> b = 9

=> a + c = 2(9)

=> a + c = 18

=> c = 18 - a (Equation 1)

Also,

a - 1, b - 1 & c + 3 are in GP

=> (b - 1)² = (a - 1) (c + 3)

=> (9 - 1)² = (a - 1)(c + 3)

=> 8² = (a - 1)(c + 3)

=> (a - 1)(c + 3) = 64 (Equation 2)

Putting Equation 1 in Equation 2, we get

(a - 1) (18 - a + 3) = 64

=> (a - 1)(21 - a) = 64

=> 21a - a² - 21 + a = 64

=> -a² + 22a -85 = 0

=> a² - 22a + 85 = 0

=> a² - 17a - 5a + 85 = 0

=> a(a - 17) -5(a - 17) = 0

=> (a - 5)(a - 17) = 0

=> a = 5 or a = 17

Since it is an decreasing AP, a should be greater than b = 9

=> a = 17

=> c = 18 - a = 18 - 17

=> c = 1

Therefore, the numbers are 17, 9 & 1