The first term of A.P. is 5, the last is 45 and their sum is 400. If the number of terms is n and d is the common differences , then `((n)/(d))` is equal to :

Options

A. 9
B. 8
C. 7
D. 6

Explanation

To solve the problem step by step, we will use the formulas related to Arithmetic Progression (A.P.).

### Step 1: Identify the given values
The first term \( a \) is given as 5, the last term \( l \) is given as 45, and the sum \( S_n \) is given as 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:
\[ S_n = \frac{n}{2} \times (a + l) \]
Substituting the known values into the formula:
\[ 400 = \frac{n}{2} \times (5 + 45) \]
This simplifies to:
\[ 400 = \frac{n}{2} \times 50 \]

### Step 3: Solve for \( n \)
Multiply both sides by 2 to eliminate the fraction:
\[ 800 = n \times 50 \]
Now, divide both sides by 50:
\[ n = \frac{800}{50} = 16 \]

### Step 4: Use the nth term formula to find the common difference \( d \)
The nth term of an A.P. is given by:
\[ l = a + (n - 1) \cdot d \]
Substituting the known values:
\[ 45 = 5 + (16 - 1) \cdot d \]
This simplifies to:
\[ 45 = 5 + 15d \]

### Step 5: Solve for \( d \)
Subtract 5 from both sides:
\[ 40 = 15d \]
Now, divide both sides by 15:
\[ d = \frac{40}{15} = \frac{8}{3} \]

### Step 6: Find the ratio \( \frac{n}{d} \)
Now that we have \( n = 16 \) and \( d = \frac{8}{3} \), we can find the ratio:
\[ \frac{n}{d} = \frac{16}{\frac{8}{3}} = 16 \times \frac{3}{8} = \frac{48}{8} = 6 \]

### Final Answer
Thus, the value of \( \frac{n}{d} \) is \( 6 \).

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