Find the first 4 terms of the recursively defined sequence. \[ a_{1}=6, a_{n+1}=1+\frac{1}{a_{n}} \] \[ a_{2}=\square \] (Simplify your answer. Type an integer, a fraction or a mixed number.) \[ a_{3}=\square \] (Simplify your answer. Type an integer, a fraction or a mixed number.) \[ a_{4}=\square \] (Simplify your answer. Type an integer, a fraction or a mixed number.)
Solution
Step 1 ：The problem is asking for the first 4 terms of a recursively defined sequence. The first term is given as 6, and the formula to find the next term is also given. We can use this formula to find the next three terms.
Step 2 ：Using the formula \(a_{n+1}=1+\frac{1}{a_{n}}\), we can find the second term \(a_{2}\) by substituting \(a_{1}=6\) into the formula, which gives us \(a_{2}=1+\frac{1}{6}=\frac{7}{6}\).
Step 3 ：Similarly, we can find the third term \(a_{3}\) by substituting \(a_{2}=\frac{7}{6}\) into the formula, which gives us \(a_{3}=1+\frac{6}{7}=\frac{13}{7}\).
Step 4 ：Finally, we can find the fourth term \(a_{4}\) by substituting \(a_{3}=\frac{13}{7}\) into the formula, which gives us \(a_{4}=1+\frac{7}{13}=\frac{20}{13}\).
Step 5 ：Final Answer: The first four terms of the sequence are \(\boxed{6}\), \(\boxed{\frac{7}{6}}\), \(\boxed{\frac{13}{7}}\), and \(\boxed{\frac{20}{13}}\).
