Question

a graphic designer wants to know the number of regions that are formed when circles overlap in a particular way. can she find a rule that describes how the number of regions increases when another circle is added to the design? how many regions would a design with 6 circles create? number of regions 1 3 7 13 21 describe the pattern. select the correct choice below and fill in the answer box to complete your choice. (type an integer or a decimal.) a. as the number of circles increases, the number of regions increases by successive multiples of each time. b. as the number of circles increases, the number of regions increases by a factor of each time.

Explanation:

Step1: Analyze the differences

Find the differences between consecutive numbers of regions:
$3 - 1=2$;
$7 - 3 = 4$;
$13 - 7=6$;
$21 - 13 = 8$.

Step2: Identify the pattern

The differences $2,4,6,8$ are consecutive even - numbers. So, as the number of circles increases, the number of regions increases by successive multiples of 2 each time.

Step3: Find the number of regions for 6 circles

The pattern for the number of regions $R(n)$ formed by $n$ circles can be found by the formula $R(n)=n^2 - n+ 1$.
For $n = 6$, we substitute into the formula:
$R(6)=6^2-6 + 1=36-6 + 1=31$.

Answer:

A. As the number of circles increases, the number of regions increases by successive multiples of 2 each time; 31