Refer to the figures. The method of successive differences can be applied to the sequence of interior regions-1,2,4,8,16,31-to find the number of regions determined by seven points on the circle. What is the nextern in this sequence? How many regions would be determined by eight points? Click here to view the first five circles Click here to view the sixth circle

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To determine the number of interior regions formed by points on a circle, we can use combinatorial methods. Specifically, when $$ n $$ points are placed on a circle and all possible chords are drawn (with the condition that no three chords intersect at the same interior point), the number of regions $$ R(n) $$ can be calculated using the formula:

$$
R(n) = \binom{n}{4} + \binom{n}{2} + 1
$$

Where:
- $$ \binom{n}{4} $$ represents the number of ways to choose 4 points (which corresponds to the intersection points of the chords).
- $$ \binom{n}{2} $$ accounts for the chords themselves.
- The "+1" accounts for the initial outer region.

Let's apply this formula to find the next terms in your sequence:

1. **For $$ n = 7 $$:**
   $$
   R(7) = \binom{7}{4} + \binom{7}{2} + 1 = 35 + 21 + 1 = 57
   $$
   
2. **For $$ n = 8 $$:**
   $$
   R(8) = \binom{8}{4} + \binom{8}{2} + 1 = 70 + 28 + 1 = 99
   $$

**Summary:**
- **Next term in the sequence (for 7 points):** **57 regions**
- **Number of regions for 8 points:** **99 regions**

Answer: The next term is 57 and eight points on the circle determine 99 interior regions.
The next term in the sequence is 57, and eight points on the circle determine 99 interior regions.

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