Q24. - Algebra 1

Question

Find the next three terms in each geometric sequence.

$32,8,2\mathrm{.....}$

Step-by-Step Solution

Answer

The next three terms in the geometric sequence are $\frac{1}{2},\frac{1}{8},\frac{1}{32}$.

Step 1. State the concept used.

In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. 

For example, the sequence 2, 6, 18, 54, ... is a geometric progression with common ratio 3.

The ration of 2^nd term and first term is called the common ratio. 

Step 2. Find the common difference.

The sequence: $32,8,2\mathrm{.....}$

Each term in a geometric sequence can be expressed in terms of the first term $a_1$ and the common ratio r. Since each succeeding term is formulated from one or more previous terms, this is a recursive formula.

First term $a_1=32$ and the common ratio is:

$$\begin{gathered} r=\frac{a_2}{a_1} \\ =\frac{8}{32} \\ =\frac{1}{4} \end{gathered}$$. 

Step 3. Find the next three terms.

In order to calculate the next three terms of the geometric sequence substitute $n=4,5,6$ and 32 for $a_1$ into the formula $a_n=a_1{r}^{n-1}$.

 

Terms	Symbol	In terms of $a_1$ and r	Numbers
Fourth term	$a_4$	$a_1\cdot r^3$	$32{\left(\frac{1}{4}\right)}^3=32\times \frac{1}{64}=\frac{1}{2}$
Fifth term	$a_5$	$a_1\cdot r^4$	$32{\left(\frac{1}{4}\right)}^4=32\times \frac{1}{256}=\frac{1}{8}$
Sixth term	$a_6$	$a_1\cdot r^5$	$32{\left(\frac{1}{4}\right)}^5=32\times \frac{1}{1024}=\frac{1}{32}$

 

Thus, next three terms of the sequence $32,8,2\mathrm{.....}$ are $\frac{1}{2},\frac{1}{8},\frac{1}{32}$.