Find a, b such that 7.2, a, b, 3 are in A.P.

Answer

To find the values of \( a \) and \( b \) such that \( 7.2, a, b, 3 \) are in Arithmetic Progression (A.P.), we can follow these steps:

### Step 1: Understand the properties of A.P.
In an A.P., the difference between consecutive terms is constant. This common difference is denoted as \( d \).

### Step 2: Set up the equations
Given the terms \( 7.2, a, b, 3 \):
- The first term \( A_1 = 7.2 \)
- The second term \( A_2 = a \)
- The third term \( A_3 = b \)
- The fourth term \( A_4 = 3 \)

From the definition of A.P., we can express the common difference \( d \) as follows:
1. \( a - 7.2 = d \) (Equation 1)
2. \( b - a = d \) (Equation 2)
3. \( 3 - b = d \) (Equation 3)

### Step 3: Equate the differences
Since all three expressions are equal to \( d \), we can set them equal to each other:
From Equations 1 and 3:
\[ a - 7.2 = 3 - b \]

From Equation 2:
\[ b - a = a - 7.2 \]

### Step 4: Solve for \( b \) in terms of \( a \)
From \( a - 7.2 = 3 - b \):
\[ b = 3 - (a - 7.2) \]
\[ b = 10.2 - a \] (Equation 4)

### Step 5: Substitute \( b \) into Equation 2
Now substitute Equation 4 into Equation 2:
\[ (10.2 - a) - a = a - 7.2 \]
\[ 10.2 - 2a = a - 7.2 \]

### Step 6: Rearrange and solve for \( a \)
Rearranging gives:
\[ 10.2 + 7.2 = 3a \]
\[ 17.4 = 3a \]
\[ a = \frac{17.4}{3} = 5.8 \]

### Step 7: Find \( b \)
Now substitute \( a = 5.8 \) back into Equation 4 to find \( b \):
\[ b = 10.2 - 5.8 \]
\[ b = 4.4 \]

### Final Answer
Thus, the values of \( a \) and \( b \) are:
\[ a = 5.8, \quad b = 4.4 \]

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