Geometric Sequences I

Problem 223: Consider the sequence $a = \{8,16,32,64,128,\dots \}$

a. What is the common ration?

b. What are the next five terms in the sequence?

Solution: A geometric sequence is different from an arithmetic sequence. In an arithmetic sequence, we look for a common difference between the terms in the sequence whereas in the geometric sequence we look for a common ratio between the terms of the sequence.

a. Note that

(1) $\begin{align*}\frac{16}{8} & = 2 \\ \frac{32}{26}& =2 \\ \frac{64}{32} & = 2 \\ \frac{128}{64} & = 2. \end{align*}$

This means that the common ratio between the terms is 2.

b. Using the common ratios we can get the next five terms.

(2) $\begin{align*} 128(2) & = 256 \\ 256(2) & = 512 \\ 512(2) & = 1024 \\ 1024(2) & = 2048 \\ 2048(2) & = 4096\end{align*}$

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