If the common difference of an AP is 10, then what is `a_(18) - a_(16)`?

Options

A. 5
B. 20
C. 25
D. 30

Answer

To solve the problem, we need to find the value of \( a_{18} - a_{16} \) given that the common difference \( d \) of the arithmetic progression (AP) is 10.

### Step-by-Step Solution:

1. **Understand the formula for the nth term of an AP:**
The nth term of an arithmetic progression can be expressed as:
\[ a_n = a + (n - 1)d \]
where \( a \) is the first term, \( n \) is the term number, and \( d \) is the common difference.

2. **Calculate \( a_{18} \):**
Using the formula for the 18th term:
\[ a_{18} = a + (18 - 1)d = a + 17d \]

3. **Calculate \( a_{16} \):**
Using the formula for the 16th term:
\[ a_{16} = a + (16 - 1)d = a + 15d \]

4. **Find \( a_{18} - a_{16} \):**
Now, we can subtract \( a_{16} \) from \( a_{18} \):
\[ a_{18} - a_{16} = (a + 17d) - (a + 15d) \]
Simplifying this gives:
\[ a_{18} - a_{16} = a + 17d - a - 15d = 17d - 15d = 2d \]

5. **Substitute the value of \( d \):**
We know from the problem that \( d = 10 \):
\[ a_{18} - a_{16} = 2d = 2 \times 10 = 20 \]

### Final Answer:
Thus, the value of \( a_{18} - a_{16} \) is \( 20 \).

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