Question

Find $a_{30}$ given that the first few terms of a geometric sequence are given by $-2,1,-\frac{1}{2},\frac{1}{4}....$

• A. $\frac{1}{2^{27}}$
• B. $\frac{1}{2^{26}}$
• C. $\frac{1}{2^{28}}$
• D. $\frac{1}{2^{29}}$

Answer 1

The correct option is C $\frac{1}{2^{28}}$

In a geometric series $a_n=a\cdot r^{n-1}$, where $a_1$ is the first term of the sequence, $r$ is the common ratio, and $n$ is the number of the term to find.

Since the first term is $-2$ and second term is $1$, therefore, the common ratio is :

$r=\frac{a_2}{a_1}=\frac{1}{-2}=-\frac{1}{2}$

Now we find $a_{30}$ given that $a=-2$, $r=-\frac{1}{2}$ and $n=30$

$a_{30}=-2{(-\frac{1}{2})}^{29}=-2(-\frac{1}{2^{29}})=\frac{2}{2^{29}}=\frac{1}{2^{28}}$