Question

A type of cell reproduces by splitting itself in half. One cell becomes 2 cells, 2 cells become 4 cells, and 4 cells become 8 cells. What type of sequence does the number of cells represent, and why? It represents an arithmetic sequence because it has a common difference of 2. It represents an arithmetic sequence because it has a common difference of 1/2. It represents a geometric sequence because it has a common ratio of 1/2. It represents a geometric sequence because it has a common ratio of 2.

Answer 1

It represents a geometric sequence because it has a common ratio of 2.

In fact, a sequence is said to be geometric if any two adjacent elements are in the same ratio. In other words, if you choose any index $n \in \mathbb{N}$ and consider the two consecutive terms $a_n$ and $a_{n+1}$, you have

$\dfrac{a_{n+1}}{a_n} = r$

no matter which index you chose. In your case, the next term in the sequence is always the double of the previous one, so the ratio between two consecutive terms is always 2, and the series is geometric with common ratio 2