To solve this problem, we can follow these steps:
1. **Determine the common difference (d)**: The common difference can be found by using the formula for the nth term of an arithmetic progression, which is given by:
\[ a_n = a_1 + (n - 1)d \]
where \( a_n \) is the nth term, \( a_1 \) is the first term, \( n \) is the term number, and \( d \) is the common difference.
Given the 10th term (16) and knowing the first term (2), we can write:
\[ a_{10} = a_1 + (10 - 1)d \]
\[ 16 = 2 + 9d \]
Solving this equation for \( d \) gives us:
\[ 16 - 2 = 9d \]
\[ 14 = 9d \]
\[ d = \frac{14}{9} \]
\[ d = \frac{14}{9} \]
2. **Find the 15th term (a_15)**: Now that we know the common difference \( d \), we can use the same formula for the nth term to find the 15th term:
\[ a_{15} = a_1 + (15 - 1)d \]
\[ a_{15} = 2 + (15 - 1) \left(\frac{14}{9}\right) \]
\[ a_{15} = 2 + 14 \times \frac{14}{9} \]
\[ a_{15} = 2 + \frac{196}{9} \]
Simplify the fraction and then add the 2:
\[ \frac{196}{9} = 21\frac{7}{9} \]
\[ a_{15} = 2 + 21\frac{7}{9} \]
\[ a_{15} = 23\frac{7}{9} \]
So, the 15th term of the arithmetic progression is \( 23\frac{7}{9} \).
