For an A.P., the sum of its terms is 60, common difference is 2 and last term is 18. Find the number of terms in the A.P.

Answer

To solve the problem step by step, we will use the information given about the arithmetic progression (A.P.) to find the number of terms.

### Step 1: Write down the given information
- Sum of the A.P. (S) = 60
- Common difference (d) = 2
- Last term (a_n) = 18

### Step 2: Use the formula for the sum of the first n terms of an A.P.
The formula for the sum of the first n terms of an A.P. is given by:
\[ S_n = \frac{n}{2} \times (a + a_n) \]
where:
- \( S_n \) is the sum of the first n terms,
- \( n \) is the number of terms,
- \( a \) is the first term,
- \( a_n \) is the last term.

Substituting the known values into the formula:
\[ 60 = \frac{n}{2} \times (a + 18) \]

### Step 3: Rearranging the equation
Multiply both sides by 2 to eliminate the fraction:
\[ 120 = n \times (a + 18) \]
Thus, we can express \( n \) as:
\[ n = \frac{120}{a + 18} \]

### Step 4: Use the formula for the nth term of an A.P.
The nth term of an A.P. is given by:
\[ a_n = a + (n - 1) \times d \]
Substituting the known values:
\[ 18 = a + (n - 1) \times 2 \]
Rearranging gives:
\[ 18 = a + 2n - 2 \]
Thus:
\[ a + 2n = 20 \quad \text{(1)} \]

### Step 5: Substitute \( n \) from step 3 into equation (1)
Substituting \( n = \frac{120}{a + 18} \) into equation (1):
\[ a + 2\left(\frac{120}{a + 18}\right) = 20 \]
Multiply through by \( a + 18 \) to eliminate the fraction:
\[ a(a + 18) + 240 = 20(a + 18) \]
Expanding both sides:
\[ a^2 + 18a + 240 = 20a + 360 \]
Rearranging gives:
\[ a^2 - 2a - 120 = 0 \]

### Step 6: Solve the quadratic equation
Now we will factor the quadratic equation:
\[ (a - 12)(a + 10) = 0 \]
Thus, the solutions for \( a \) are:
\[ a = 12 \quad \text{or} \quad a = -10 \]

### Step 7: Find the number of terms \( n \)
Now, we will find \( n \) for both values of \( a \).

**For \( a = 12 \)**:
\[ n = \frac{120}{12 + 18} = \frac{120}{30} = 4 \]

**For \( a = -10 \)**:
\[ n = \frac{120}{-10 + 18} = \frac{120}{8} = 15 \]

### Conclusion
The number of terms in the A.P. can be either 4 or 15.