Find five numbers in A.P. whose sum is `12 1/2 ` and the ratio of the first to the last terms is 2:3.

Answer

To find five numbers in an arithmetic progression (A.P.) whose sum is \(12 \frac{1}{2}\) and the ratio of the first to the last terms is \(2:3\), we can follow these steps:

### Step 1: Define the Terms
Let's denote the five numbers in A.P. as:
- First term: \(A - 2D\)
- Second term: \(A - D\)
- Third term: \(A\)
- Fourth term: \(A + D\)
- Fifth term: \(A + 2D\)

### Step 2: Set Up the Sum Equation
The sum of these five terms can be expressed as:
\[ (A - 2D) + (A - D) + A + (A + D) + (A + 2D) = 12 \frac{1}{2} \]
Converting \(12 \frac{1}{2}\) to an improper fraction gives us:
\[ 12 \frac{1}{2} = \frac{25}{2} \]
Thus, we can rewrite the equation as:
\[ 5A = \frac{25}{2} \]

### Step 3: Solve for \(A\)
To find \(A\), we divide both sides by 5:
\[ A = \frac{25}{2 \times 5} = \frac{25}{10} = \frac{5}{2} \]

### Step 4: Set Up the Ratio Equation
We know that the ratio of the first term to the last term is \(2:3\):
\[ \frac{A - 2D}{A + 2D} = \frac{2}{3} \]
Cross-multiplying gives:
\[ 3(A - 2D) = 2(A + 2D) \]

### Step 5: Expand and Rearrange
Expanding both sides:
\[ 3A - 6D = 2A + 4D \]
Rearranging the equation gives:
\[ 3A - 2A = 6D + 4D \implies A = 10D \]

### Step 6: Substitute \(A\) to Find \(D\)
Now substituting \(A = \frac{5}{2}\) into \(A = 10D\):
\[ \frac{5}{2} = 10D \implies D = \frac{5}{2 \times 10} = \frac{5}{20} = \frac{1}{4} \]

### Step 7: Calculate the Five Terms
Now we can find the five terms:
1. First term:
\[ A - 2D = \frac{5}{2} - 2 \times \frac{1}{4} = \frac{5}{2} - \frac{2}{4} = \frac{5}{2} - \frac{1}{2} = \frac{4}{2} = 2 \]
2. Second term:
\[ A - D = \frac{5}{2} - \frac{1}{4} = \frac{10}{4} - \frac{1}{4} = \frac{9}{4} \]
3. Third term:
\[ A = \frac{5}{2} \]
4. Fourth term:
\[ A + D = \frac{5}{2} + \frac{1}{4} = \frac{10}{4} + \frac{1}{4} = \frac{11}{4} \]
5. Fifth term:
\[ A + 2D = \frac{5}{2} + 2 \times \frac{1}{4} = \frac{5}{2} + \frac{2}{4} = \frac{5}{2} + \frac{1}{2} = \frac{6}{2} = 3 \]

### Final Result
The five numbers in A.P. are:
\[ 2, \frac{9}{4}, \frac{5}{2}, \frac{11}{4}, 3 \]