The first term of an A.P. is 5 , the last term is 45 and the sum is 400. Find the number of terms and the common difference .

Answer

To solve the problem, we need to find the number of terms (n) and the common difference (d) of the arithmetic progression (A.P.) given the first term (a), last term (l), and the sum of the A.P. (S).

### Step-by-Step Solution:

1. **Identify the given values:**
- First term (a) = 5
- Last term (l) = 45
- Sum of the A.P. (S) = 400

2. **Use the formula for the sum of the first n terms of an A.P.:**
\[ S = \frac{n}{2} (a + l) \]
Substituting the known values:
\[ 400 = \frac{n}{2} (5 + 45) \]
Simplifying further:
\[ 400 = \frac{n}{2} \times 50 \]
\[ 400 = 25n \]

3. **Solve for n:**
\[ n = \frac{400}{25} = 16 \]

4. **Now, use the formula for the nth term of an A.P.:**
The nth term (l) can also be expressed as:
\[ l = a + (n - 1) d \]
Substituting the known values:
\[ 45 = 5 + (16 - 1) d \]
Simplifying this:
\[ 45 = 5 + 15d \]
\[ 45 - 5 = 15d \]
\[ 40 = 15d \]

5. **Solve for d:**
\[ d = \frac{40}{15} = \frac{8}{3} \]

### Final Answers:
- Number of terms (n) = 16
- Common difference (d) = \(\frac{8}{3}\)