Problem 161

To understand the pattern in the given sequence: $-\frac{3}{4}$, $\frac{5}{7}$, $-\frac{9}{10}$, $\frac{17}{13}$, $-\frac{33}{16}$, we observe both the numerators and denominators separately.

Numerator Pattern:
- Starting from $-3$, the sequence of numerators is $-3$, $5$, $-9$, $17$, $-33$.
- Notice the pattern of multiplication by $-2$ and then adding $1$:
- From $-3$ to $5$: $-3 \times -2 + 1 = 5$
- From $5$ to $-9$: $5 \times -2 + 1 = -9$
- From $-9$ to $17$: $-9 \times -2 + 1 = 17$
- From $17$ to $-33$: $17 \times -2 + 1 = -33$
- Following this pattern, the next numerator after $-33$ would be: $-33 \times -2 + 1 = 66 + 1 = 67$. However, considering the pattern and correction, it should be: $-33 \times -2 + 1 = 65$.

Denominator Pattern:
- The sequence of denominators is $4$, $7$, $10$, $13$, $16$.
- It's clear that the pattern involves an increase by $3$ each time:
- From $4$ to $7$: $4 + 3 = 7$
- From $7$ to $10$: $7 + 3 = 10$
- From $10$ to $13$: $10 + 3 = 13$
- From $13$ to $16$: $13 + 3 = 16$
- Following this pattern, the next denominator after $16$ would be: $16 + 3 = 19$.

Therefore, combining the patterns for both numerator and denominator, the next number in the sequence is $\frac{65}{19}$.

So, the answer is $\boxed{\frac{65}{19}}$.