Problem 92 - College Algebra

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 results are \(\frac{-1}{2}\) for n=1, \(\frac{1}{8}\) for n=2, \(\frac{-1}{26}\) for n=3, and \(\frac{1}{80}\) for n=4.

Step 1: Substitute n=1

Start with the value n=1. Substituting this into the expression, we get \(\frac{(-1)^{1}}{3^{1}-1}=\frac{-1}{2}\).

Step 2: Substitute n=2

Next is value n=2. Make the substitution and compute the answer, which yields \(\frac{(-1)^{2}}{3^{2}-1}=\frac{1}{8}\).

Step 3: Substitute n=3

Follow the same process with n=3. The result is \(\frac{(-1)^{3}}{3^{3}-1}=\frac{-1}{26}\).

Step 4: Substitute n=4

Lastly, evaluate the expression for n=4. This gives \(\frac{(-1)^{4}}{3^{4}-1}=\frac{1}{80}\).