Problem: Find the value of a:b:c if a, b, c are in arithmetic progression and b-a, c-b, a are in geometric progression.

Solution:

Step 1: Understanding the Problem
We are given that a, b, c are in arithmetic progression and b-a, c-b, a are in geometric progression. We need to find the value of a:b:c.

Step 2: Using formulas for arithmetic and geometric progressions
Let the common difference of the arithmetic progression be d. Then we have:
b = a + d
c = b + d = a + 2d

Let the common ratio of the geometric progression be r. Then we have:
b - a = r^1
c - b = r^2
a = r^3

Step 3: Solving for a, b, c
Substituting the values of b and c in terms of a and d, we get:
a + d - a = r^1
a + 2d - (a + d) = r^2
a = r^3

Simplifying, we get:
d = r - 1
d(r + 1) = r^2
a = r^3

Solving these equations, we get:
r = (1 + sqrt(5))/2 (ignoring the negative solution as r is a ratio)
d = sqrt(5)
a = (3 + sqrt(5))/2
b = a + d = (5 + sqrt(5))/2
c = b + d = (7 + sqrt(5))/2

Therefore, the value of a:b:c is:
(3 + sqrt(5))/2 : (5 + sqrt(5))/2 : (7 + sqrt(5))/2

Step 4: Answer
The value of a:b:c is (3 + sqrt(5))/2 : (5 + sqrt(5))/2 : (7 + sqrt(5))/2.

Conclusion: We have solved the problem by using the formulas for arithmetic and geometric progressions. The final answer is (3 + sqrt(5))/2 : (5 + sqrt(5))/2 : (7 + sqrt(5))/2.

Given :- a,b,c are in A.P and b-a,c-b,a are in G.P To Find :- a:b:c Solution :- As a,b,c are in A.P first term =a common difference =d then a=a,b=a+d,c=a+2d then b-a=a+d-a=d c-b=a+2d-a-d=d and also b-a,c-b,a are in G.P then d^2=a×d=ad d^2-ad=0 then d(d-a)=0 then d=0 or a=d then a:b:c=1:1:1 or 1:2:3