Formulas Powerpoint (level 6)
... Because the difference is two you must multiply the number by two, ...
... Because the difference is two you must multiply the number by two, ...
Sequences Revision
... Position to term rules It is important to be able to find the number in a sequence from its position in that sequence. Example. If we want to find the 100th term in the sequence ...
... Position to term rules It is important to be able to find the number in a sequence from its position in that sequence. Example. If we want to find the 100th term in the sequence ...
Full text
... Each distinct value of terms in E and Q occurs exactly twice, except the first and third values, which occur only once. Observe that no three consecutive terms of E are increasing or decreasing, since the values alternate in magnitude; the same is true of Q, since the primes form an increasing seque ...
... Each distinct value of terms in E and Q occurs exactly twice, except the first and third values, which occur only once. Observe that no three consecutive terms of E are increasing or decreasing, since the values alternate in magnitude; the same is true of Q, since the primes form an increasing seque ...
1. Prove the second part of De Morgan’s Laws, namely... A ∪ B = A ∩ B.
... 1. Prove the second part of De Morgan’s Laws, namely for sets A and B A ∪ B = A ∩ B. ...
... 1. Prove the second part of De Morgan’s Laws, namely for sets A and B A ∪ B = A ∩ B. ...
Practice questions for Exam 1
... 11. For each of the following, determine if a set is a subset, proper subset, or equal to the other set, or state that none of these properties can be inferred. (a) What can we say for the sets A and B if we know that A ∪ B = A? (b) What can we say for the sets A and B if we know that A − B = A? ...
... 11. For each of the following, determine if a set is a subset, proper subset, or equal to the other set, or state that none of these properties can be inferred. (a) What can we say for the sets A and B if we know that A ∪ B = A? (b) What can we say for the sets A and B if we know that A − B = A? ...
Collatz conjecture
The Collatz conjecture is a conjecture in mathematics named after Lothar Collatz, who first proposed it in 1937. The conjecture is also known as the 3n + 1 conjecture, the Ulam conjecture (after Stanisław Ulam), Kakutani's problem (after Shizuo Kakutani), the Thwaites conjecture (after Sir Bryan Thwaites), Hasse's algorithm (after Helmut Hasse), or the Syracuse problem; the sequence of numbers involved is referred to as the hailstone sequence or hailstone numbers (because the values are usually subject to multiple descents and ascents like hailstones in a cloud), or as wondrous numbers.Take any natural number n. If n is even, divide it by 2 to get n / 2. If n is odd, multiply it by 3 and add 1 to obtain 3n + 1. Repeat the process (which has been called ""Half Or Triple Plus One"", or HOTPO) indefinitely. The conjecture is that no matter what number you start with, you will always eventually reach 1. The property has also been called oneness.Paul Erdős said about the Collatz conjecture: ""Mathematics may not be ready for such problems."" He also offered $500 for its solution.