Question

you said stage 20 will have $2^{10}$ circles. how many times as many circles will there be at stage 21 compared to stage 20? these are the same. twice as many three times as many five times as many

Explanation:

Step1: Assume a pattern of circle - growth

Let's assume the number of circles at stage $n$ follows a geometric - sequence pattern. If the number of circles at stage $n$ is $a_n$, and the common ratio of the geometric sequence is $r$. Then $a_{n + 1}=r\times a_n$.

Step2: Analyze the relationship between consecutive stages

In a geometric - sequence, if the number of circles at stage 20 is $a_{20}=2^{10}$, and the number of circles at stage 21 is $a_{21}$, and assuming a common ratio of 2 (a common exponential - growth pattern), then $a_{21}=2\times a_{20}$.

Answer:

Twice as many