Questions: 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 next term in this sequence? How many regions would be determined by eight points? The next term in the sequence is . regions would be determined by eight points.

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 next term in this sequence? How many regions would be determined by eight points?

The next term in the sequence is .
regions would be determined by eight points.

Transcript text: 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 next term in this sequence? How many regions would be determined by eight points? The next term in the sequence is $\square$ . $\square$ regions would be determined by eight points.

Solution

Solution Steps

Step 1: Identify the Given Sequence

The given sequence of interior regions is $ 1, 2, 4, 8, 16, 31 $.

Step 2: Calculate Successive Differences

We calculate the successive differences of the sequence:

First differences: $ 2 - 1 = 1 $, $ 4 - 2 = 2 $, $ 8 - 4 = 4 $, $ 16 - 8 = 8 $, $ 31 - 16 = 15 $ resulting in $ 1, 2, 4, 8, 15 $.
Second differences: $ 2 - 1 = 1 $, $ 4 - 2 = 2 $, $ 8 - 4 = 4 $, $ 15 - 8 = 7 $ resulting in $ 1, 2, 4, 7 $.
Third differences: $ 2 - 1 = 1 $, $ 4 - 2 = 2 $, $ 7 - 4 = 3 $ resulting in $ 1, 2, 3 $.
Fourth differences: $ 2 - 1 = 1 $, $ 3 - 2 = 1 $ resulting in $ 1, 1 $.
Fifth differences: $ 1 - 1 = 0 $ resulting in $ 0 $.

Step 3: Determine the Next Term

Since the last non-zero difference is $ 15 $ and the next difference is $ 0 $, we conclude that the next term in the sequence remains $ 31 $. Thus, the next term is $ 31 $.

Step 4: Calculate Regions for Eight Points

To find the number of regions determined by eight points, we use the formula for the number of regions formed by $ n $ points on a circle: \[ R(n) = \frac{n(n-1)}{2} + 1 \] Substituting $ n = 8 $: \[ R(8) = \frac{8 \cdot 7}{2} + 1 = 28 + 1 = 29 \] Thus, the number of regions determined by eight points is $ 29 $.