Question

Calculate the missing terms of the geometric sequence ..., 2048, ?, ?, ?,8,.... Select all that apply.A. 1,538, 1,028, 518B. 1,534, 1,020,514C. 512, 128, 32D. -512,128, -32E -1,534, 1,020,- 514OF -1,538,1,028,- 518

Answer 1

Answer:

• C. 512, 128, 32

,

• D. -512,128, -32

Explanation:

Given the geometric sequence with missing terms below:

$...,2048,?,?,?,8,...$

In order to find the missing terms, consider the finite partial sequence below:

$2048,?,?,?,8$

• The first term, a1 = 2048

,

• The fifth term, a5 = 8

The nth term of a geometric sequence is calculated using the formula:

$a_n=a_1r^{n-1}$

Thus:

$\begin{gathered} a_5=a_1r^{5-1} \\ \implies8=2048r^4 \\ \text{Divide both sides by 2048} \\ \frac{8}{2048}=\frac{2048r^4}{2048} \\ r^4=\frac{1}{256} \\ r=\sqrt[4]{\frac{1}{256}} \\ r=\pm\frac{1}{4} \end{gathered}$

The common ratio is either 1/4 or -1/4.

When the common ratio is 1/4, the missing terms in the geometric sequence are:

$\begin{gathered} 2048\times\frac{1}{4}=512 \\ 512\times\frac{1}{4}=128 \\ 128\times\frac{1}{4}=32 \end{gathered}$

When the common ratio is -1/4, the missing terms in the geometric sequence are:

$\begin{gathered} 2048\times-\frac{1}{4}=-512 \\ -512\times-\frac{1}{4}=128 \\ 128\times-\frac{1}{4}=-32 \end{gathered}$

Options C and D are correct.