Find the general term, an, of the given geometric sequence.
-$\frac { 1 } { 9 } , \frac { 5 } { 9 } , \frac { 25 } { 9 } , \ldots$
A) $a _ { n } = \frac { 1 } { 9 } + 5 ( n - 1 )$
B) $a _ { n } = \frac { 1 } { 9 } + \frac { 4 } { 9 } ( n - 1 )$
C) $a _ { n } = \left( \frac { 1 } { 9 } \right) ^ { n - 1 } + \frac { 4 } { 9 }$
D) $a _ { n } = \frac { 1 } { 9 } ( 5 ) ^ { n - 1 }$

Question

Quizplus · Accepted Answer

In a geometric sequence, each term is obtained by multiplying the previous term by a constant factor called the common ratio. To find the common ratio, we divide any term by the previous term.

Let's use the second and first terms to find the common ratio:
$$\frac{5/9}{1/9}=\frac{5}{1}=5$$

So, the common ratio is 5.

To find the general term, we can use the formula: $a_n=a_1\cdot r^{n-1}$, where $a_1$ is the first term and $r$ is the common ratio. The first term is $a_1=1/9$, and the common ratio is $r=5$. Substituting these values into the formula, we get:

$$a_n=\frac{1}{9}\cdot5^{n-1}$$

Therefore, the correct choice is D.

Find the General Term, An, of the Given Geometric Sequence 19,59,259,…\frac { 1 } { 9 } , \frac { 5 } { 9 } , \frac { 25 } { 9 } , \ldots91​,95​,925​,…

Find the General Term, An, of the Given Geometric Sequence $\frac { 1 } { 9 } , \frac { 5 } { 9 } , \frac { 25 } { 9 } , \ldots$