Problem 59 - Algebra and Trigonometry

Question

Will help you prepare for the material covered in the first section of the next chapter. Evaluate \(\frac{(-1)^{n}}{3^{n}-1}\) for \(n=1,2,3,\) and 4

Step-by-Step Solution

Answer

The evaluated values of the sequence \(\frac{(-1)^{n}}{3^{n}-1}\) for \(n=1,2,3,4\) are -1/2, 1/8, -1/26, 1/80 respectively.

Step 1: Evaluate the Sequence for \(n=1\)

We'll plug \(n=1\) into the sequence. So, \( \frac{(-1)^{1}}{3^{1}-1} = \frac{-1}{2} .\)

Step 2: Evaluate the Sequence for \(n=2\)

Similarly, we'll plug \(n=2\) into the sequence. That gives us \( \frac{(-1)^{2}}{3^{2}-1} = \frac{1}{8} . \)

Step 3: Evaluate the Sequence for \(n=3\)

Plugging \(n=3\) into the sequence, we get \( \frac{(-1)^{3}}{3^{3}-1} = \frac{-1}{26} .\)

Step 4: Evaluate the Sequence for \(n=4\)

Finally, for \(n=4\), the sequence gives \(\frac{(-1)^{4}}{3^{4}-1} = \frac{1}{80} .\)