Let's go through the geometric sequences shown in the image and solve for the next three terms for each one.

1. Sequence 1: $1, \frac{1}{2}, \frac{1}{4}, \ldots$

   • To find the common ratio ($r$), divide the second term by the first: $r = \frac{\frac{1}{2}}{1} = \frac{1}{2}$
   • Now, multiply each term by $\frac{1}{2}$ to get the next terms: $\frac{1}{4} \times \frac{1}{2} = \frac{1}{8}$ $\frac{1}{8} \times \frac{1}{2} = \frac{1}{16}$ $\frac{1}{16} \times \frac{1}{2} = \frac{1}{32}$
   • So, the next three terms are $\frac{1}{8}, \frac{1}{16}, \frac{1}{32}$.

2. Sequence 2: $100, 20, 4, \ldots$

   • Find the common ratio: $r = \frac{20}{100} = \frac{1}{5}$
   • Multiply the terms by $\frac{1}{5}$: $4 \times \frac{1}{5} = \frac{4}{5}$ $\frac{4}{5} \times \frac{1}{5} = \frac{4}{25}$ $\frac{4}{25} \times \frac{1}{5} = \frac{4}{125}$
   • The next three terms are $\frac{4}{5}, \frac{4}{25}, \frac{4}{125}$.

3. Sequence 3: $\frac{8}{3}, \frac{4}{3}, \frac{2}{3}, \ldots$

   • Find the common ratio: $r = \frac{\frac{4}{3}}{\frac{8}{3}} = \frac{1}{2}$
   • Multiply the terms by $\frac{1}{2}$: $\frac{2}{3} \times \frac{1}{2} = \frac{1}{3}$ $\frac{1}{3} \times \frac{1}{2} = \frac{1}{6}$ $\frac{1}{6} \times \frac{1}{2} = \frac{1}{12}$
   • The next three terms are $\frac{1}{3}, \frac{1}{6}, \frac{1}{12}$.

4. Sequence 4: $-1, 2, -4, \ldots$

   • Find the common ratio: $r = \frac{2}{-1} = -2$
   • Multiply the terms by $-2$: $-4 \times (-2) = 8$ $8 \times (-2) = -16$ $-16 \times (-2) = 32$
   • The next three terms are $8, -16, 32$.

Would you like any more details on these solutions?

Related Questions:

1. How do you find the common ratio in a geometric sequence?
2. What is the difference between geometric and arithmetic sequences?
3. How do you find the sum of a geometric series?
4. How do negative ratios affect the behavior of a geometric sequence?
5. How can you apply geometric sequences in real-life scenarios?

Tip:

Always double-check the common ratio when multiplying to find the next terms—it ensures your sequence remains correct!