Al has 75 days to master discrete mathematics. He decides to study at least one hour every day, but no more than a total of 125 hours. Assume Al always studies in one hour units. Show there must be a sequence of consecutive days during which he studies exactly 24 hours.

Avery Stewart 2022-07-18 Answered
Al has 75 days to master discrete mathematics. He decides to study at least one hour every day, but no more than a total of 125 hours. Assume Al always studies in one hour units. Show there must be a sequence of consecutive days during which he studies exactly 24 hours.
You can still ask an expert for help

Want to know more about Discrete math?

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

iljovskint
Answered 2022-07-19 Author has 18 answers
Step 1
Let a k be the number of hours of work Al has done after k days. Then { a 1 , a 2 , a 3 , , a 75 } is an increasing sequence of distinct positive integers since Al does at least one hour of work each day. Observe that 1 a k 125 since Al does at most 125 hours of work over the 75 days.
Let b k = a k + 24. Then the sequence { b 1 , b 2 , b 3 , , b 75 } is also an increasing sequence of distinct positive integers. Observe that 1 + 24 = 25 b k 149 = 125 + 24.
Step 2
Now consider the union of the two sequences. It consists of 150 numbers that are at least 1 and at most 149. Thus, two of them must be the same. Hence, b k = a k + 24 = a j for some j,k. Thus, a j a k = 24, so Al does exactly 24 hours of work from day a k + 1 to day a j .
This is a clever application of the Pigeonhole Principle that forced me to consult my combinatorics notes.
Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2021-08-15
How many elements are in the set { 0, { { 0 } }?
asked 2021-08-18
Discrete Mathematics Basics
1) Determine whether the relation R on the set of all Web pages is reflexive, symmetric, antisymmetric, and/or transitive, where (a,b)R if and only if
I) everyone who has visited Web page a has also visited Web page b.
II) there are no common links found on both Web page a and Web page b.
III) there is at least one common link on Web page a and Web page b.
asked 2020-11-09
Use proof by Contradiction to prove that the sum of an irrational number and a rational number is irrational.
asked 2021-08-02
Suppose that A is the set of sophomores at your school and B is the set of students in discrete mathematics at your school. Express each of these sets in terms of A and B.
a) the set of sophomores taking discrete mathematics in your school
b) the set of sophomores at your school who are not taking discrete mathematics
c) the set of students at your school who either are sophomores or are taking discrete mathematics
Use these symbols:
asked 2022-04-19

Solve the laplace inverse:

Solve the differential equation y''+2y'+17y=0, y(0) and y'(0)=12

asked 2021-07-30
Prove that discrete math by induction that for all integers n1
i=1ni2=n(n+1)(2n+1)6
asked 2022-06-14
Total number of subsets in a set
I was reading about subsets, in that, the article suggests the total number of subsets in a set is 2 n , where n is the number of elements in the set. For example - { 1 , 2 , 3 , 4 , 5 } the total number of subsets is 32 because n is 5 and 2 5 is 32 by multiplicative principle.
But the multiplicative principle is that if m events can happen in n ways then the possible outcomes are m × n. So in the subsets problem if every element has 2 possibilities of it being in set or not being in set why is it not 2 × 5 and 2 5 ? I know that the 2 5 is correct but not able to visualize it.

New questions

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question