Problem 12 - College Algebra

Question

Write the first four terms of each sequence whose general term is given. $$a_{n}=\frac{(-1)^{n+1}}{2^{n}+1}$$

Step-by-Step Solution

Answer

The first four terms of the given sequence are \(\frac{1}{3}\), \(-\frac{1}{5}\), \(\frac{1}{9}\), \(-\frac{1}{17}\).

Step 1: Substitute n=1 into the general term

Substitute \(n=1\) into the general term: \[a_{1}=\frac{(-1)^{1+1}}{2^{1}+1}= \frac{(-1)^2}{2+1} = \frac{1}{3}\]

Step 2: Substitute n=2 into the general term

Substitute \(n=2\) into the general term: \[a_{2}=\frac{(-1)^{2+1}}{2^{2}+1}= \frac{-1}{4+1} = -\frac{1}{5}\]

Step 3: Substitute n=3 into the general term

Substitute \(n=3\) into the general term: \[a_{3}=\frac{(-1)^{3+1}}{2^{3}+1}= \frac{1}{8+1} = \frac{1}{9}\]

Step 4: Substitute n=4 into the general term

Substitute \(n=4\) into the general term: \[a_{4}=\frac{(-1)^{4+1}}{2^{4}+1}= \frac{-1}{16+1} = -\frac{1}{17}\]