The sum of `25` terms of an `A.P.`, whose all the terms are natural numbers, lies between `1900` and `2000` and its `9^(th)` term is `55`. Then the first term of the `A.P.` is

Options

A. `5`
B. `6`
C. `7`
D. `8`

Explanation

To solve the problem step by step, we will follow the information given in the question regarding the arithmetic progression (A.P.).

### Step 1: Understand the given information
We know that:
- The 9th term of the A.P. is 55.
- The sum of the first 25 terms lies between 1900 and 2000.

### Step 2: Use the formula for the nth term of an A.P.
The nth term of an A.P. is given by:
\[ a_n = a + (n - 1)d \]
For the 9th term:
\[ a + 8d = 55 \quad \text{(1)} \]

### Step 3: Use the formula for the sum of the first n terms of an A.P.
The sum of the first n terms (S_n) is given by:
\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]
For the first 25 terms:
\[ S_{25} = \frac{25}{2} \times (2a + 24d) \]
We know that \(1900 < S_{25} < 2000\). Therefore:
\[ 1900 < \frac{25}{2} \times (2a + 24d) < 2000 \]

### Step 4: Simplify the inequality
Multiply the entire inequality by 2 to eliminate the fraction:
\[ 3800 < 25(2a + 24d) < 4000 \]
Now divide by 25:
\[ 152 < 2a + 24d < 160 \quad \text{(2)} \]

### Step 5: Substitute \(d\) from equation (1) into equation (2)
From equation (1), we can express \(a\):
\[ a = 55 - 8d \]
Substituting this into equation (2):
\[ 152 < 2(55 - 8d) + 24d < 160 \]
This simplifies to:
\[ 152 < 110 - 16d + 24d < 160 \]
\[ 152 < 110 + 8d < 160 \]

### Step 6: Solve the inequalities
Now, we will solve the inequalities:
1. For the left inequality:
\[ 152 - 110 < 8d \implies 42 < 8d \implies d > 5.25 \]
2. For the right inequality:
\[ 8d < 160 - 110 \implies 8d < 50 \implies d < 6.25 \]

### Step 7: Determine the integer value of \(d\)
Since \(d\) must be a natural number, the only integer value that satisfies \(5.25 < d < 6.25\) is:
\[ d = 6 \]

### Step 8: Substitute \(d\) back to find \(a\)
Now substitute \(d = 6\) back into equation (1):
\[ a + 8(6) = 55 \]
\[ a + 48 = 55 \]
\[ a = 55 - 48 = 7 \]

### Conclusion
The first term of the A.P. is:
\[ \boxed{7} \]