Exercise 1. (a) Using full induction to n ∈ N, prove that (a) Let XXXXXXXXXXDetermine (. Exercise 2. Resolve the recurring relationship Using formal power series. (Afterwards, check your answer by...

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Exercise 1. (a) Using full induction to n ∈ N, prove that (a) Let Determine (. Exercise 2. Resolve the recurring relationship Using formal power series. (Afterwards, check your answer by calculating in two ways.) Exercise 3. (a) Let be the number of partitions of a number n. Explain that According to the table below, . (You can also calculate that table yourself using Wolfram Alpha; for example, enter this: series product [1 / (1-x ^ k), k = 1 ... 10].) (b) List the partitions (among these 42 partitions) that consist of odd parts (such as 10 = 3 + 3 + 1 + 1 + 1 + 1). (c) Also list the partitions (among these 42 partitions) that consist of unequal parts (such as 10 = 5 + 3 + 2). You should have found two equally long lists. Leonhard Euler showed in 1748 that for all natural numbers the number of partitions with odd parts is always equal to the number of partitions with unequal parts. (d) Show using formal power series that the number of partitions of a number in odd parts is indeed equal to the number of partitions of in unequal parts.