University/Undergrad Math Spiral Review: Sequences, Logic, and Set Theory

Question 1

Determine if the sequence 4, 14, 24, 34 is arithmetic or geometric. If arithmetic, find the equation for the nth term.

Answer: The sequence is arithmetic with a common difference of 10.

Question 2

Determine if the sequence 4, 24, 144, 864 is arithmetic or geometric. If geometric, provide the geometric equation for the nth term.

Answer: The nth term can be described by the equation a_n = 4r^(n-1) where r = 6. The first term is 4, the common ratio (r) is 6.

Question 3

A student claimed that 4, 24, 44, and 64 all have a remainder of 0 when divided by four, so all numbers ending in 4 must have 0 remainder when divided by 4. How do you respond?

Answer: A counterexample shows that not all numbers ending in 4 are divisible by 4, as seen with 14. Hence, their statement is not valid.

Question 4

The first windmill has 5 matchstick squares, the second has 9, and the third has 13. How many matchstick squares are in the fifth windmill? Provide an equation for the number of matchstick squares in the nth windmill.

Answer: In the fifth windmill, there are 21 matchstick squares. The equation is S_n = 4n + 1.

Question 5

Which of the following is a negation of 'All chairs are gray.'?

Answer: The negation is "There exists at least one chair that is not gray."

Question 6

Which of the following is the inverse of 'All birds are red'?

Answer: The inverse is 'If something is not a bird, then it is not red.'

Question 7

Complete the truth table for p ∨ q and for q → ~p.

Answer: The truth table is completed as follows. p ∨ q: | p | q | p∨q | |:-:|:-:|:--:| | T | T | T | | T | F | T | | F | T | T | | F | F | F | q → ~p: | q | ~p | q→~p | |:-:|:-:|:--:| | T | F | F | | T | T | T | | F | F | T | | F | T | T |

Question 8

Consider the following: Hypotheses: All TAMU students take two math classes. Janie is not a TAMU student. Draw and label an Euler diagram.

Answer: The diagram shows that Janie is outside the TAMU student circle and hence no conclusions can be made about her math classes taken. She may or may not take two classes.

Question 9

Using sets U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, V = {1, 2, 3, 4, 5}, X = {2, 4, 6, 8}, identify V ∩ X, V ∪ X, V̅, and list the proper subset(s) of V.

Answer: V ∩ X = {2, 4}; V ∪ X = {1, 2, 3, 4, 5, 6, 8}; V̅ = {6, 7, 8, 9}; Proper subset(s) of V: {1}, {2}, {3}, {4}, {5}, {1, 2}, {1, 3}, ..., {1, 2, 3, 4, 5}.

Question 10

If n(A) = 6, n(B) = 8, n(C) = 10. What is the greatest number of elements in A ∩ B ∩ C? What is the least number of elements in A ∪ B ∪ C?

Answer: The greatest number of elements in A ∩ B ∩ C can be 6. The least number of elements in A ∪ B ∪ C can be 18.

Question 11

Write set R in roster notation if R = {3x + 1|x ∈ W, x < 5}.

Answer: R = {4, 7, 10, 13, 16} based on values of x: 0, 1, 2, 3, 4.

Question 12

In an organization with 500 members, 220 take mathematics, 80 take both mathematics and English, and 45 take neither mathematics nor English. How many members of this organization take English?

Answer: 235 members take English.

Notes

This review covers key concepts in sequences, logical operations, and set theory to reinforce understanding and application of mathematical principles.