Week 5 notes

The primary goal of this course is to become familiar with some of the basic mathematical ideas used in programming.

Homework 4

p 108: 3abc. Also give the total count. 3c is not a stars-and-bars, but would be if the letters had to be in alphabetical order.

p 144: 2, 4, 5

Levin 1.6: Derangements

1.6.4: Derangements

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2.1 Sequences

Tower of Hanoi

Factorials, triangle numbers

Recursively defined sequences

Example 2.1.2

Python example like 2.1.2 third sequence (but note differences); compare to sixth sequence

import math

phi=(1+math.sqrt(5))/2

pho=(1-math.sqrt(5))/2

def s(n):
    return (pow(phi,n)-pow(pho,n))/math.sqrt(5)

for i in range(20):
    print(i,'\t',s(i))

Recursive (or inductive) definitions

Example 2.1.3

def a(n):
    if n==0: return 3
    if n==1: return 4
    return 2*a(n-1) - a(n-2)    

But there is a much simpler way to calculate a(n)!

p 140:

• Square numbers
• Triangle numbers
• powers of 2
• Fibonacci numbers

Example 2.1.4: first step is to identify the patterns

2.1.5: summation notation

2.1.6: more summation

2.2: Arithmetic and Geometric Sequences

p 148: Investigate!

Example 2.2.1: arithmetic sequences

Example 2.2.2: geometric sequences

Sums of arithmetic and geometric sequences

1 + 2 + 3 + ... + 100

Example 2.2.4

Example 2.2.5

Example 2.2.6: fix

1+2+4+...+512

Example 2.2.7

Example 2.2.8

Example 2.2.9

    S = 0.464646.... When we subtract from 100S, we get +46 on the right, but there is no minus term. Why?

2.3: Polynomial Fitting

Friday:

mathwithbaddrawings.com/2017/02/08/how-to-tell-a-mathematician-you-love-them

Example 2.2.6 on Levin p 154

Let b_k = 1+3k. Then the sum

a_n = 2 + 1 + 4 + 7 + 10 + · · · + (1 + 3(n−1))

can be rewritten as

a_n =2 + b₀ + b₁ + ... + b_n-1.

Written this way it is clearer that there are n terms of the form b_k here, ranging from 0 to (n-1).

Homework 4 questions

Proof by induction: 1 + 2 + ... + N = N(N+1)/2

Find 1² + 2² + 3² + ... + N².

2.4: Recurrence Relations

Investigate!, page 167

def a(n):
    if n== 0: return 1
    if n==1: return 2
    return 5*a(n-1) - 6*a(n-2)

def b(n):
    if n== 0: return 1
    if n==1: return 3
    return 5*b(n-1) - 6*b(n-2)

def c(n):
    if n== 0: return 1
    if n==1: return 4
    return 5*c(n-1) - 6*c(n-2)

Example 2.4.2, p 168

Example 2.4.3: a_n = a_n-1 + n

Example 2.4.4: same problem, alternative approach

Example 2.4.5: introduce a factor: a_n = 3*a_n-1 + 2. Make a guess?

def a(n):
    if n== 0: return 1
    return 3*a(n-1) + 2

Characteristic roots

Example 2.4.6

Example at start of chapter: a_n = 5*a_n-1 - 6*a_n-2

Fibonacci example

2.5: Mathematical Induction

Postage in Investigate!

Example 2.5.1: triangle numbers

Example 2.5.2: 6ⁿ-1 is divisible by 5

Example 2.5.3: n² < 2ⁿ for n>=5

Warning: Canadians

Euclidean algorithm theorem: for integers a and b<a, we can find q and r so a=qb+r