Suppose there are three numbers a,b,c (in increasing order) such that a,b,c are in AP and a²,b²,c² are in GP. If a + b + c = 3/2 then what is the value of a?

Question

Answer

A, b, c are in AP .

Therefore a = b-d and c= b+d

Given, a+b+c = 3/2

b= 1/2

Hence, a=1/2-d, b=1/2,c=1/2+d

Given, a²,b²,c² are in GP.

(b²)²= a²c²

Solving the above equation, we get,

(1/2-d)(1/2+d)= (±1/4)

Solve for d, and we get d= 1/√2 (d greater than equal to 0)

a = 1/2 - 1/√2

Answer

Answer

**Given Information:**

- There are three numbers a, b, c in increasing order.
- The numbers a, b, c are in arithmetic progression (AP).
- The squares of the numbers a², b², c² are in geometric progression (GP).
- The product of the numbers a, b, c is equal to 3/2.

**Solution:**

Let's solve this problem step by step:

**Step 1: Understanding the Arithmetic Progression (AP)**

In an arithmetic progression (AP), the difference between any two consecutive terms is constant. Let the common difference be 'd'.

Therefore, we can express the terms of AP as:

a = a
b = a + d
c = a + 2d

**Step 2: Understanding the Geometric Progression (GP)**

In a geometric progression (GP), the ratio between any two consecutive terms is constant. Let the common ratio be 'r'.

Therefore, we can express the squares of the terms of GP as:

a² = ar²⁰
b² = ar²¹
c² = ar²²

**Step 3: Finding the Common Ratio (r)**

Since a², b², c² are in GP, we can find the common ratio 'r' by dividing any two consecutive terms:

b² / a² = (a + d)² / a² = (a² + 2ad + d²) / a²
c² / b² = (a + 2d)² / (a + d)² = (a² + 4ad + 4d²) / (a² + 2ad + d²)

Since a², b², c² are in GP, the ratio b² / a² = c² / b².

Therefore, we can equate the two expressions:

(a² + 2ad + d²) / a² = (a² + 4ad + 4d²) / (a² + 2ad + d²)

Cross-multiplying and simplifying, we get:

a⁴ + 2a³d + a²d² = a⁴ + 4a³d + 4a²d²

2a³d = 3a²d²
2a² = 3d

**Step 4: Finding the Value of 'a'**

We are given that the product of a, b, c is equal to 3/2:

a * (a + d) * (a + 2d) = 3/2

Substituting the value of 'd' from Step 3, we get:

a * (a + 2a²/3) * (a + 4a²/3) = 3/2

Simplifying the expression, we get:

2a⁴ + 12a⁴ + 16a⁴ = 9

30a⁴ = 9

a⁴ = 3/10

Taking the fourth root on both sides, we get:

a = ∛(3/10)

Therefore, the value of 'a' is ∛(3/10), which is the solution to the given problem.

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