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Math 243 Foundations
Assignment Number 8
Due: Friday, October 30
Problem List
4.3.16, 5.1.13
1. LATEX (4.3.16) For the sequence a1 , a2 , . . . , an , . . . , assume that a1 = 1, and that for each natural
number n,
an+1 = an + n · n!.
(a) Compute n! for the first 10 natural numbers.
(b) Compute an for the first 10 natural numbers.
(c) Make a conjecture about a formula for an in terms of n that does not involve a summation or a
recursion.
(d) Prove your conjecture in Part (c).
2. (5.1.13) We can extend the idea of consecutive integers (See Exercise (1) in Section 3.5) to represent
four consecutive integers as m, m + 1, m + 2, and m + 3, where m is an integer. There are other ways
to represent four consecutive integers. For example, if k ∈ Z, then k − 1, k, k + 1, and k + 2 are four
consecutive integers.
(a) Prove that for each n ∈ Z, n is the sum of four consecutive integers if and only if n ≡ 2 (mod 4).
(b) Use set builder notation or the roster method to specify the set of integers that are the sum of
four consecutive integers.
(c) Specify the set of all natural numbers that can be written as the sum of four consecutive natural
numbers.
(d) Prove that for each n ∈ Z, n is the sum of eight consecutive integers if and only if n ≡ 4 (mod 8).
(e) Use set builder notation or the roster method to specify the set of integers that are the sum of
eight consecutive integers.
(f) Specify the set of all natural numbers that can be written as the sum of eight consecutive natural
numbers.