Find the number of terms in the A.P. 7, 10, 13, .........., 31.

Answer

To find the number of terms in the arithmetic progression (A.P.) 7, 10, 13, ..., 31, we can follow these steps:

### Step 1: Identify the first term and common difference
The first term \( a \) of the A.P. is 7. The common difference \( d \) can be calculated by subtracting the first term from the second term:
\[ d = 10 - 7 = 3 \]

### Step 2: Use the formula for the nth term of an A.P.
The formula for the nth term \( A_n \) of an A.P. is given by:
\[ A_n = a + (n - 1) \cdot d \]
where:
- \( A_n \) is the nth term,
- \( a \) is the first term,
- \( n \) is the number of terms,
- \( d \) is the common difference.

### Step 3: Set up the equation for the last term
In this case, the last term \( A_n \) is 31. We can set up the equation as follows:
\[ 31 = 7 + (n - 1) \cdot 3 \]

### Step 4: Simplify the equation
Now, we will simplify the equation:
\[ 31 = 7 + 3(n - 1) \]
Subtract 7 from both sides:
\[ 31 - 7 = 3(n - 1) \]
\[ 24 = 3(n - 1) \]

### Step 5: Solve for \( n \)
Now, divide both sides by 3:
\[ 8 = n - 1 \]
Add 1 to both sides:
\[ n = 9 \]

### Conclusion
Thus, the number of terms in the A.P. 7, 10, 13, ..., 31 is \( n = 9 \).

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