Question

Suppose the sequence \(\left\\{a_{n}\right\\}\) is defined by the explicit formula \(a_{n}=1 / n,\) for \(n=1,2,3, \ldots .\) Write out the first five terms of the sequence.

Short answer

Answer: The first five terms are \(1, 0.5, 0.333, 0.25, 0.2\).

Step-by-step solution

1. Find the first term of the sequence#a_1\(

To find the first term of the sequence, we'll substitute \)n=1\( into the formula \)a_n = \frac{1}{n}$. So: $$ a_1 = \frac{1}{1} = 1 $$

2. Find the second term of the sequence\(a_2\)

Now, we'll substitute \(n=2\) into the formula \(a_n = \frac{1}{n}\). So: $$ a_2 = \frac{1}{2} = 0.5 $$

3. Find the third term of the sequence\(a_3\)

Next, we'll substitute \(n=3\) into the formula \(a_n = \frac{1}{n}\). So: $$ a_3 = \frac{1}{3} \approx 0.333 $$

4. Find the fourth term of the sequence\(a_4\)

Now we'll substitute \(n=4\) into the formula \(a_n = \frac{1}{n}\). So: $$ a_4 = \frac{1}{4} = 0.25 $$

5. Find the fifth term of the sequence\(a_5\)

Finally, we'll substitute \(n=5\) into the formula \(a_n = \frac{1}{n}\). So: $$ a_5 = \frac{1}{5} = 0.2 $$

Result: First five terms of the sequence

Following the steps outlined above, we have found the first five terms of the sequence \(\left\\{a_{n}\right\\}\): $$ a_1 = 1, \quad a_2 = 0.5, \quad a_3 \approx 0.333, \quad a_4 = 0.25, \quad a_5 = 0.2 $$