For this week's brain potion we will explore some various problems involving sequences. Caution, you'll probably want a calculator for some of these!

For each of the sequences below: (i) Describe the pattern and calculate the first few numbers. (ii) What number do you think the sequence is approaching?

- Consider the following pattern of numbers $$\displaystyle \frac{4}{1}, \frac{4}{1} - \frac{4}{3}, \frac{4}{1} - \frac{4}{3} + \frac{4}{5}, \frac{4}{1} - \frac{4}{3} + \frac{4}{5} - \frac{4}{7}, \ldots$$
- Consider now this pattern of numbers: $$\displaystyle 2\times \frac{2}{1}, 2\times \frac{2}{1}\frac{2}{3}, 2\times \frac{2}{1}\frac{2}{3}\frac{4}{3}, 2\times \frac{2}{1}\frac{2}{3}\frac{4}{3}\frac{4}{5}, 2\times \frac{2}{1}\frac{2}{3}\frac{4}{3}\frac{4}{5}\frac{6}{5},\ldots$$
- Here's another pattern: $$3, 3 + \frac{4}{2\cdot3\cdot4}, 3 + \frac{4}{2\cdot3\cdot4} - \frac{4}{4\cdot5\cdot6}, 3 + \frac{4}{2\cdot3\cdot4} - \frac{4}{4\cdot5\cdot6} + \frac{4}{6\cdot7\cdot8}, \ldots$$
- One last pattern: $$3, 3 + 1^2, 3 + \frac{1^2}{6 + 3^2}, 3 + \frac{1^2}{6 + \dfrac{3^2}{6+5^2}}, 3 + \frac{1^2}{6 + \dfrac{3^2}{6+\dfrac{5^2}{6+7^2}}}, \ldots$$

As a challenge, use Excel/Google sheets or write a program to help calculate the values of these sequences!

Please click here to view and participate in this weeks challenge! Good Luck!