Write the expression for the general (or nth)termrm, $\mathbf { a } _ { \mathbf { n } } ,$ , of the geometric sequence.
-$- \frac { 1 } { 3 } , - \frac { 1 } { 27 } , - \frac { 1 } { 243 } , - \frac { 1 } { 2187 } , \ldots$
A) $a _ { n } = \left( - \frac { 1 } { 3 } \right) ^ { n - 1 } + \frac { 8 } { 27 }$
B) $a _ { n } = - \frac { 1 } { 3 } + \frac { 8 } { 27 } ( n - 1 )$
C) $a _ { n } = - \frac { 1 } { 3 } + \frac { 1 } { 9 } ( n - 1 )$
D) $a _ { n } = - \frac { 1 } { 3 } \left( \frac { 1 } { 9 } \right) ^ { n - 1 }$

Question

Quizplus · Accepted Answer

The given sequence is a geometric sequence where each term is obtained by multiplying the previous term by $-\frac{1}{9}$ (since $-\frac{1}{27} \div -\frac{1}{3} = -\frac{1}{9}$, and so on for the subsequent terms). The first term $a_1$ is $-\frac{1}{3}$, and the common ratio $r$ is $-\frac{1}{9}$. The formula for the nth term of a geometric sequence is $a_n = a_1 \cdot r^{n-1}$. Substituting $a_1 = -\frac{1}{3}$ and $r = -\frac{1}{9}$ into this formula, we get $a_n = -\frac{1}{3} \cdot \left(-\frac{1}{9}\right)^{n-1}$, which simplifies to $a_n = -\frac{1}{3} \left(\frac{1}{9}\right)^{n-1}$, matching option D.

Write the Expression for the General (Or Nth)termrm an,\mathbf { a } _ { \mathbf { n } } ,an​,

Write the Expression for the General (Or Nth)termrm $\mathbf { a } _ { \mathbf { n } } ,$