Which term of the A.P. 102+108+…is 210?

Explanation

To find which term of the arithmetic progression (A.P.) 102, 108, ... is equal to 210, we can follow these steps:

### Step 1: Identify the first term (a) and the common difference (d)
The first term \( a \) of the A.P. is given as:
\[ a = 102 \]

The second term is:
\[ a_2 = 108 \]

To find the common difference \( d \):
\[ d = a_2 - a_1 = 108 - 102 = 6 \]

### 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:
\[ a_n = a + (n - 1) \cdot d \]

### Step 3: Set up the equation for the nth term
We need to find \( n \) such that:
\[ a_n = 210 \]

Substituting the known values into the formula:
\[ 210 = 102 + (n - 1) \cdot 6 \]

### Step 4: Solve for n
First, isolate \( (n - 1) \cdot 6 \):
\[ 210 - 102 = (n - 1) \cdot 6 \]
\[ 108 = (n - 1) \cdot 6 \]

Next, divide both sides by 6:
\[ n - 1 = \frac{108}{6} \]
\[ n - 1 = 18 \]

Now, add 1 to both sides to find \( n \):
\[ n = 18 + 1 \]
\[ n = 19 \]

### Conclusion
The term of the A.P. that equals 210 is the 19th term.