MA 410 Spring 2025 Homework 1
Due as indicated for each problem.
All solutions must be submitted on the Moodle web site
for the class at
wolfware.ncsu.edu.
You may upload a photo of your handwritten solution or
a file of your typed solution.
Note my office hours on my
schedule.
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Due Thurs January 23, 11:59pm.
Prove by induction that for the n-th Chebyshev-2 Polynomial Un(x) in the variable x,
defined by the linear recurrence U0 = 1, U1(x) = 2x,
Un+2(x) = 2xUn+1(x) – Un(x) for all n ≥ 0
one has
(y–1/y) Un( (y + 1/y)/2 ) = yn+1 –
y–n–1 for all n ≥ 0, y ≠ 0.
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Due Thurs January 23, 11:59pm.
Consider the Catalan numbers as defined
Cn = binomial(2n,n) / (n+1)
in ENT, §1.2, Problem 10, page 12 for all n ≥ 0.
Prove that Cn = binomial(2n,n) – binomial(2n,n–1) for
all n ≥ 1, thus showing that the Cn are integers.
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Due Thurs Jan 30, 11:59pm.
ENT, §2.2, Problem 23, page 26:
if a divides bc show that a divides gcd(a,b)gcd(a,c).
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Due Thurs Jan 30, 11:59pm.
Please compute integers x,y,z such that 63x + 70y + 90z = 1.
Please show your work, computing a solution to 63x + 70y = 7 and
7w + 90z = 1 first. Note: you only need to give one triple.
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Due Tues February 4, 11:59pm.
Prove for the extended Euclidean algorithm described in class:
sk tk+1 – sk+1 tk
= (–1)k+1
for all –1 ≤ k ≤ n –1.