1. The problem is to find 5 sequences that approach the number 1, with some sequences increasing towards 1 and others decreasing towards 1. 2. A sequence \(a_n\) approaches 1 if \(\lim_{n \to \infty} a_n = 1\). 3. For increasing sequences approaching 1, each term is less than the next and all terms are less than 1 but get closer to 1. 4. For decreasing sequences approaching 1, each term is greater than the next and all terms are greater than 1 but get closer to 1. 5. Here are examples of 5 increasing sequences approaching 1: | n | Sequence 1: \(1 - \frac{1}{n+1}\) | Sequence 2: \(1 - \frac{1}{2^n}\) | Sequence 3: \(1 - \frac{1}{n^2}\) | Sequence 4: \(1 - \frac{1}{3^n}\) | Sequence 5: \(1 - \frac{1}{n!}\) | |---|-------------------------------|----------------------------|--------------------------|----------------------------|--------------------------| | 1 | 0.5 | 0.5 | 0 | 0.6667 | 0 | | 2 | 0.6667 | 0.75 | 0.75 | 0.8889 | 0.5 | | 3 | 0.75 | 0.875 | 0.8889 | 0.96296 | 0.8333 | | 4 | 0.8 | 0.9375 | 0.9375 | 0.98765 | 0.9583 | | 5 | 0.8333 | 0.96875 | 0.96 | 0.99537 | 0.9917 | 6. Here are examples of 5 decreasing sequences approaching 1: | n | Sequence 1: \(1 + \frac{1}{n}\) | Sequence 2: \(1 + \frac{1}{2^n}\) | Sequence 3: \(1 + \frac{1}{n^2}\) | Sequence 4: \(1 + \frac{1}{3^n}\) | Sequence 5: \(1 + \frac{1}{n!}\) | |---|-------------------------------|----------------------------|--------------------------|----------------------------|--------------------------| | 1 | 2 | 1.5 | 2 | 1.3333 | 2 | | 2 | 1.5 | 1.25 | 1.25 | 1.1111 | 1.5 | | 3 | 1.3333 | 1.125 | 1.1111 | 1.0370 | 1.1667 | | 4 | 1.25 | 1.0625 | 1.0625 | 1.0123 | 1.0417 | | 5 | 1.2 | 1.03125 | 1.04 | 1.0041 | 1.0083 | 7. Each sequence gets closer to 1 as \(n\) increases, with increasing sequences approaching 1 from below and decreasing sequences from above.