All the terms of an AP are natural numbers and the sum of the first 20 terms is greater than 1072 and lss than 1162.If the sixth term is 32, then

Options

A. first term is 7
B. first term is 12
C. common difference is 4
D. common difference is 5

Explanation

To solve the problem step by step, we will use the properties of an arithmetic progression (AP) and the given conditions.

### Step 1: Define the terms of the AP
Let the first term of the AP be \( a \) and the common difference be \( d \). The \( n \)-th term of an AP can be expressed as:
\[ a_n = a + (n-1)d \]
Given that the 6th term is 32, we can write:
\[ a + 5d = 32 \quad \text{(1)} \]

### Step 2: Use the formula for the sum of the first \( n \) terms
The sum of the first \( n \) terms of an AP is given by:
\[ S_n = \frac{n}{2} \times (2a + (n-1)d) \]
For the first 20 terms, we have:
\[ S_{20} = \frac{20}{2} \times (2a + 19d) = 10 \times (2a + 19d) = 20a + 190d \]
We know that:
\[ 1072 < S_{20} < 1162 \quad \text{(2)} \]

### Step 3: Substitute \( a \) from equation (1) into the sum
From equation (1), we can express \( a \) in terms of \( d \):
\[ a = 32 - 5d \]
Substituting this into the sum equation:
\[ S_{20} = 20(32 - 5d) + 190d = 640 - 100d + 190d = 640 + 90d \]
Thus, we have:
\[ 1072 < 640 + 90d < 1162 \quad \text{(3)} \]

### Step 4: Solve the inequalities
Now we will solve the inequalities in (3).

**First Inequality:**
\[ 640 + 90d > 1072 \]
Subtract 640 from both sides:
\[ 90d > 432 \]
Dividing by 90:
\[ d > \frac{432}{90} = 4.8 \quad \text{(4)} \]

**Second Inequality:**
\[ 640 + 90d < 1162 \]
Subtract 640 from both sides:
\[ 90d < 522 \]
Dividing by 90:
\[ d < \frac{522}{90} = 5.8 \quad \text{(5)} \]

### Step 5: Determine the integer value of \( d \)
From inequalities (4) and (5), we have:
\[ 4.8 < d < 5.8 \]
The only natural number that satisfies this condition is:
\[ d = 5 \]

### Step 6: Find the first term \( a \)
Now substitute \( d = 5 \) back into equation (1) to find \( a \):
\[ a = 32 - 5 \times 5 = 32 - 25 = 7 \]

### Conclusion
The first term \( a \) is 7, and the common difference \( d \) is 5.

### Final Answer:
- First term \( a = 7 \)
- Common difference \( d = 5 \)

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