Midterm 2 Review Answers
... The error can be thought of as occurring either in line 2 or line 3. The mistake is due to improperly grouping sub-expressions, i.e. being lazy about the use of parentheses. In line 2 the sets (A ∪ B) and (A ∪ C) should be grouped together by yet another pair of parentheses due to the ∩ between them ...
... The error can be thought of as occurring either in line 2 or line 3. The mistake is due to improperly grouping sub-expressions, i.e. being lazy about the use of parentheses. In line 2 the sets (A ∪ B) and (A ∪ C) should be grouped together by yet another pair of parentheses due to the ∩ between them ...
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... if the number of sides of a regular polygon is of the form 2k Fm1 . . . Fmr , where k 0, r 0, and Fmi are distinct Fermat primes, then this polygon can be constructed by ruler and compass. The converse statement was established later by Wantzel in [8]. There exist many necessary and sufficient con ...
... if the number of sides of a regular polygon is of the form 2k Fm1 . . . Fmr , where k 0, r 0, and Fmi are distinct Fermat primes, then this polygon can be constructed by ruler and compass. The converse statement was established later by Wantzel in [8]. There exist many necessary and sufficient con ...
N - HKOI
... G(x) = ((5 + sqrt(5)) / 10) / (a-x)+((5 - sqrt(5)) / 10) / (b-x) = -(sqrt(5) / 5) / (1- x/a) + (sqrt(5) / 5) / (1- x/b) • Note that 1 + rx + r2x2 +. . . = 1 / (1 - rx) G(x) = (sqrt(5) / 5)(-1-x/a-x2/a2-...+1+x/b+x2/b2+...) • By Uniqueness, Fn = (sqrt(5) / 5)(-1/an + 1/bn) ...
... G(x) = ((5 + sqrt(5)) / 10) / (a-x)+((5 - sqrt(5)) / 10) / (b-x) = -(sqrt(5) / 5) / (1- x/a) + (sqrt(5) / 5) / (1- x/b) • Note that 1 + rx + r2x2 +. . . = 1 / (1 - rx) G(x) = (sqrt(5) / 5)(-1-x/a-x2/a2-...+1+x/b+x2/b2+...) • By Uniqueness, Fn = (sqrt(5) / 5)(-1/an + 1/bn) ...
Recitation #3 – Discussion on solutons
... 2. A sequence of numbers is monotonic increasing if the values are in sorted order. For example, 1, 1, 3, 4, 9 is a monotonic sequence, but 1, 3, 2, 4, 9 is not. Design and implement a program VerifyMonoIncreasing.java that determines whether its input sequence of integers is in monotonic increasing ...
... 2. A sequence of numbers is monotonic increasing if the values are in sorted order. For example, 1, 1, 3, 4, 9 is a monotonic sequence, but 1, 3, 2, 4, 9 is not. Design and implement a program VerifyMonoIncreasing.java that determines whether its input sequence of integers is in monotonic increasing ...
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.