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 step by step, we will use the formulas related to Arithmetic Progressions (A.P.).

### Step 1: Identify the given values
- First term (A) = 5
- Last term (L) = 45
- Sum of the terms (S) = 400

### 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:
\[ S_n = \frac{n}{2} (A + L) \]
Substituting the known values into the formula:
\[ 400 = \frac{n}{2} (5 + 45) \]
\[ 400 = \frac{n}{2} \times 50 \]

### Step 3: Simplify the equation
Multiply both sides by 2 to eliminate the fraction:
\[ 800 = n \times 50 \]

### Step 4: Solve for n
Now, divide both sides by 50:
\[ n = \frac{800}{50} = 16 \]
So, the number of terms (n) is 16.

### Step 5: Use the formula for the last term of an A.P.
The formula for the last term of an A.P. is given by:
\[ L = A + (n - 1) \cdot d \]
Substituting the known values:
\[ 45 = 5 + (16 - 1) \cdot d \]
\[ 45 = 5 + 15d \]

### Step 6: Simplify and solve for d
Subtract 5 from both sides:
\[ 40 = 15d \]
Now, divide both sides by 15:
\[ d = \frac{40}{15} = \frac{8}{3} \]

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

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