Question

Find the next three terms in each geometric sequence.

$\mathbf{3}\mathbf{,}\mathbf{9}\mathbf{,}\mathbf{27}\mathbf{,}\mathbf{.}\mathbf{....}$

Short answer

The next three terms in the geometric sequence are$\mathbf{81}\mathbf{,}\mathbf{243}\mathbf{,}\mathbf{729}$.

Step-by-step solution

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 $2,6,18,54,\dots ..$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 ratio.

The sequence: $3,9,27,\mathrm{.....}$

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 $a_1=3$and the common ratio is:

$\begin{array}{c}r=\frac{a_2}{a_1}\\ =\frac{9}{3}\\ =3\end{array}$

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 3 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	$a_4$	$a_1\cdot r^3$	$3{\left(3\right)}^3=3^4=81$
Fifth term	$a_5$	$a_1\cdot r^4$	$3{\left(3\right)}^4=3^5=243$
Sixth term	$a_6$	$a_1\cdot r^5$	$3{\left(3\right)}^5=3^6=729$

Thus, next three terms of the sequence $3,9,27,\mathrm{.....}$ are $81,243,729$.