If `t_(n)={{:(n^(2)", when n is even"),(n^(2)+1", when n is odd"):}`
find `t_(15)-t_(10)`

Options

A. 116
B. 126
C. 106
D. 226

Answer

To solve the problem, we need to find \( t_{15} - t_{10} \) using the given definitions of \( t_n \):

1. **Identify \( t_{15} \)**:
- Since \( 15 \) is an odd number, we use the formula for odd \( n \):
\[ t_n = n^2 + 1 \]
- Therefore, for \( n = 15 \):
\[ t_{15} = 15^2 + 1 \]
- Calculate \( 15^2 \):
\[ 15^2 = 225 \]
- Now add 1:
\[ t_{15} = 225 + 1 = 226 \]

2. **Identify \( t_{10} \)**:
- Since \( 10 \) is an even number, we use the formula for even \( n \):
\[ t_n = n^2 \]
- Therefore, for \( n = 10 \):
\[ t_{10} = 10^2 \]
- Calculate \( 10^2 \):
\[ 10^2 = 100 \]
- So, \( t_{10} = 100 \).

3. **Calculate \( t_{15} - t_{10} \)**:
- Now we subtract \( t_{10} \) from \( t_{15} \):
\[ t_{15} - t_{10} = 226 - 100 \]
- Perform the subtraction:
\[ t_{15} - t_{10} = 126 \]

Thus, the final answer is:
\[ t_{15} - t_{10} = 126 \]