Tagged Questions
1
vote
2answers
49 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
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1answer
56 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
$$\sum_{r=1}^{k-1} r^n= \frac{P_{n+1}(k)-P_{n+1}(0)}{n+1}$$
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1
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1answer
87 views
Sequences with induction and proving. Polynomial and rational functions
$1.$We define a sequence of rational number {$a_n$} by putting $$a_1 =3,\;\text{ and}\;\; a_{n+1} = 4 - \frac{2}{a_n} \text{ for all}\; n \in \mathbb{R}.\;\text{ Put}\;\; \alpha = 2 + \sqrt{2}.$$
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0
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2answers
80 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
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2answers
90 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
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3answers
344 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
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2answers
451 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 ...
