In an A.P., the last term is 28 and the sum of all the 9 terms of the A.P. is 144. Find the first term.

Answer

To find the first term of the arithmetic progression (A.P.) where the last term is 28 and the sum of the first 9 terms is 144, we can follow these steps:

### Step 1: Write down the known values
We know:
- The number of terms (n) = 9
- The last term (A9) = 28
- The sum of the first 9 terms (S9) = 144

### Step 2: Use the formula for the sum of an A.P.
The formula for the sum of the first n terms of an A.P. is given by:
\[ S_n = \frac{n}{2} \times (A + A_n) \]
where:
- \( S_n \) is the sum of the first n terms,
- \( A \) is the first term,
- \( A_n \) is the last term (or nth term).

For our case, we can substitute the known values:
\[ S_9 = \frac{9}{2} \times (A + A_9) \]
Substituting \( S_9 = 144 \) and \( A_9 = 28 \):
\[ 144 = \frac{9}{2} \times (A + 28) \]

### Step 3: Solve for A
To eliminate the fraction, we can multiply both sides by 2:
\[ 288 = 9 \times (A + 28) \]
Now, divide both sides by 9:
\[ 32 = A + 28 \]
Next, subtract 28 from both sides to isolate A:
\[ A = 32 - 28 \]
\[ A = 4 \]

### Conclusion
The first term of the A.P. is \( A = 4 \).

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