Solve the problem.
-Find the nth term of the following geometric sequence: $1 , - \frac { 1 } { \pi } , \frac { 1 } { \pi ^ { 2 } } , \ldots$
A) $\frac { ( 7 ) ^ { n - 1 } } { \pi ^ { n } }$
B) $\frac { ( 1 ) ^ { n } } { \pi ^ { n } }$
C) $\frac { ( 1 ) ^ { n } } { \pi ^ { n - 1 } }$
D) $\frac { ( 1 ) ^ { n - 1 } } { \pi ^ { n - 1 } }$

Question

Quizplus · Accepted Answer

The sequence is a geometric sequence with the first term $a = 1$ and common ratio $r = -\frac{1}{\pi}$. The nth term of a geometric sequence is given by $a \cdot r^{n-1}$. Substituting $a = 1$ and $r = -\frac{1}{\pi}$, we get the nth term as $1 \cdot \left(-\frac{1}{\pi}\right)^{n-1}$. Since $(-1)^{n-1}$ is equivalent to $1^{n-1}$ in terms of its effect on the sign of the term (it will either be 1 or -1, which does not affect the absolute value of the term), and considering the power of $\pi$, the correct expression for the nth term simplifies to $\frac{(1)^{n-1}}{\pi^{n-1}}$, which matches option D.

Solve the Problem 1,−1π,1π2,…1 , - \frac { 1 } { \pi } , \frac { 1 } { \pi ^ { 2 } } , \ldots1,−π1​,π21​,…

Solve the Problem $1 , - \frac { 1 } { \pi } , \frac { 1 } { \pi ^ { 2 } } , \ldots$