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CS313K: Logic, Sets and Functions
Fall 2010
Problem Set 1: Infinite Sequences
The infinite sequence of numbers A 1 , A 2 ,... is defined by the condition:
An =
{ n + 1, if n is even, n − 1 , otherwise.
1.1. Calculate the first 6 members of this sequence:
n 1 2 3 4 5 6
An
Find a single formula for An that works for all numbers, both even and odd.
1.2. Find all values of n for which An < 5.
1.3. Sequence A contains every integer that is greater than 1. True or false?
1.4. Number 10 occurs in sequence A two times. True or false?
By Bn we denote the sum of all numbers from 1 to n:
Bn = 1 + 2 + · · · + n.
Cn stands for the sum of the squares of these numbers, and Dn for the sum of their cubes: Cn = 1^2 + 2^2 + · · · + n^2 ,
Dn = 1^3 + 2^3 + · · · + n^3.
Numbers Bn, Cn, Dn can be also described using “sigma notation.” For instance, the formula
D 100 = 1^3 + 2^3 +... + 100^3 ,
written in sigma notation, looks like this:
D 100 =
(^100) ∑
i=
i^3.
The letter i is the index variable in this expression, and the numbers 1 and 100 are the lower and upper bounds. (We could have used any other variable instead of i, for instance j.) More generally,
Bn =
∑^ n
i=
i, Cn =
∑^ n
i=
i^2 , Dn =
∑^ n
i=
i^3.
1.5. Calculate the first 5 members of sequence B. Find a simple formula for Bn that uses neither dots nor sigma notation.
1.6. Calculate the first 5 members of sequence D. Can you guess what a simple formula for Dn may be like?
1.7. Calculate the first 5 members of sequence C. Can you guess what a simple formula for Cn may be like?
By En we denote the number of ways to choose two elements out of n. For instance, E 4 = 6, because a 4-element set {p, q, r, s} has 6 subsets consisting of two elements each:
{p, q}, {p, r}, {p, s}, {q, r}, {q, s}, {r, s}.
1.8. Calculate the first 5 members of sequence E. Find a formula for En.
By Fn we denote the number of parts into which a plane is divided by n lines taken at random. For instance, F 3 = 7, because 3 lines taken at random divide the plane into 7 parts: a triangle and 6 infinite regions.
1.9. Calculate the first 5 members of sequence F. Find a formula for Fn.
