1
vote
1answer
32 views

Sum of nth powers and generalized polynomial sum

So this is a 2-part question (both parts I believe are closely related): How exactly does on express the sum $$\sum_{i=0}^{k}{i^n} = Q(n,k)$$ in a closed form For arbitrary positive integers ...
3
votes
3answers
234 views

A polynomial with integer coefficient

I'm struggling with this question: Suppose $P(x)$ is a polynomial with integer coefficients such that non of the values $P(1),...,P(2010)$ is divisible by $2010$. Prove that $P(n)\neq 0$ for all ...
0
votes
3answers
284 views

Prove $x^n-1$ is divisible by $x-1$ by induction

Prove that for all natural number $x$ and $n$, $x^n - 1$ is divisible by $x-1$. So here's my thoughts: it is true for $n=1$, then I want to prove that it is also true for $n-1$ then I use long ...
-1
votes
1answer
83 views

Revisited: Binomial Theorem: An Inductive Proof

I'm asked to use the fact that $\begin{pmatrix}n\\r\end{pmatrix}+\begin{pmatrix}n\\r-1\end{pmatrix}=\begin{pmatrix}n+1\\r\end{pmatrix}$ to show, by induction, that ...
0
votes
3answers
77 views

What are the coefficients of the polynomial inductively defined as $f_1=(x-2)^2\,\,\,;\,\,\,f_{n+1}=(f_n-2)^2$?

Let $\{f_n(x)\}_{n\in \Bbb N}$ be a sequence of polynomials defined inductively as $$\begin{matrix} f_1(x) & = & (x-2)^2 & \\ f_{n+1}(x)& = & (f_n(x)-2)^2, &\text{ for all ...
1
vote
2answers
90 views

How can be done by the method of mathematical induction?

We are given that $P(x+1)-P(x)=2x+1$ We also know that $P(0)=1$ We want to prove that $P(2004)=(2004)^2 +1$ Can someone explain how can be solved with mathematical induction? Thank you in advance!
1
vote
1answer
83 views

Bernoulli polynomials, Apostol

Define Bernoulli polynomials as: $P_0(x)=1$, $P'_n(x)=nP_{n-1}(x)$, $\int_0^1P_n(x)=0$ if $n\geq1$ Need to prove that for $n\geq2$ we have $$\sum_{r=1}^{k-1} r^n= \frac{P_{n+1}(k)-P_{n+1}(0)}{n+1}$$ ...
1
vote
1answer
150 views

Sequences with induction and proving. Polynomial and rational functions

$1.$We define a sequence of rational number {$a_n$} by putting $$a_1 =3,\;\text{ and}\;\; a_{n+1} = 4 - \frac{2}{a_n} \text{ for all}\; n \in \mathbb{R}.\;\text{ Put}\;\; \alpha = 2 + \sqrt{2}.$$ ...
0
votes
2answers
111 views

Quartic (degree 4) polynomial complex number problem

Can you find a quartic (degree 4) polynomial $p(x) = ax^4+bx^3+cx^2+dx+e$ with real coefficients $a$, $b$, $c$, $d$, $e$ whose roots are precisely $x=5$, $x=-2$, $x=3$ and $x=1+i$ ? Guys please help ...
1
vote
2answers
93 views

Induction (concerning $1+z+\dots+z^n$) and follow up question

I am doing a review of stuff from earlier in the semester and I can't prove this by induction: Use induction on $n$ to verify that $1+x+\cdots+z^n= \frac{1-z^{n+1}}{1-z}$ (for $z\not=1)$. Use this ...
4
votes
3answers
365 views

Question about a recursively defined function

Problem. Let $(f_n)_{n=1}^\infty$ be a sequence of functions $f_n\colon [-1,\infty)^n\to\mathbb{R}$ that are recursively defined in the following way: $$f_1(x_1)=1+x_1,$$ $$f_n(x_1,\ldots,x_n) = ...
3
votes
2answers
984 views

Proof of the polynomial division algorithm

The theorem which I am referring to states: for any $f, g$ there exist $q, r$ such that $f(x)=g(x)q(x)+r(x)$ with the degree of $r$ less than the degree of $g$ if $g$ is monic. The book I am using ...