# Tagged Questions

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### Prove a sum formula by induction

I am to prove through induction that $$\sum_{k=1}^n (2k-1)^2 = \frac{n(2n-1)(2n+1)}{3}$$ And well, my method seems to be working, but I get stuck when I'm nearly done. First I prove the formula work ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
Let $\{f_n(x)\}_{n\in \Bbb N}$ be a sequence of polynomials defined inductively as $$\begin{matrix} f_1(x) & = & (x-2)^2 & \\ f_{n+1}(x)& = & (f_n(x)-2)^2, &\text{ for all ... 2answers 99 views ### How can be done by the method of mathematical induction? We are given that P(x+1)-P(x)=2x+1 We also know that P(0)=1 We want to prove that P(2004)=(2004)^2 +1 Can someone explain how can be solved with mathematical induction? Thank you in advance! 1answer 83 views ### Bernoulli polynomials, Apostol Define Bernoulli polynomials as: P_0(x)=1, P'_n(x)=nP_{n-1}(x), \int_0^1P_n(x)=0 if n\geq1 Need to prove that for n\geq2 we have$$\sum_{r=1}^{k-1} r^n= \frac{P_{n+1}(k)-P_{n+1}(0)}{n+1}$$... 1answer 153 views ### Sequences with induction and proving. Polynomial and rational functions 1.We define a sequence of rational number {a_n} by putting$$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}.$$... 2answers 112 views ### Quartic (degree 4) polynomial complex number problem Can you find a quartic (degree 4) polynomial p(x) = ax^4+bx^3+cx^2+dx+e with real coefficients a, b, c, d, e whose roots are precisely x=5, x=-2, x=3 and x=1+i ? Guys please help ... 2answers 93 views ### Induction (concerning 1+z+\dots+z^n) and follow up question I am doing a review of stuff from earlier in the semester and I can't prove this by induction: Use induction on n to verify that 1+x+\cdots+z^n= \frac{1-z^{n+1}}{1-z} (for z\not=1). Use this ... 3answers 365 views ### Question about a recursively defined function Problem. Let (f_n)_{n=1}^\infty be a sequence of functions f_n\colon [-1,\infty)^n\to\mathbb{R} that are recursively defined in the following way:$$f_1(x_1)=1+x_1,f_n(x_1,\ldots,x_n) = ...
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 ...