Stay ahead of learning milestones! Enroll in a class over the summer!

G
Topic
First Poster
Last Poster
k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following upcoming events:
[list][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
[*]May 19th, 4:30pm PT/7:30pm ET, What's Next After Beast Academy?, designed for students finishing Beast Academy and ready for Prealgebra 1.
[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
Our full course list for upcoming classes is below:
All classes run 7:30pm-8:45pm ET/4:30pm - 5:45pm PT unless otherwise noted.

Introductory: Grades 5-10

Prealgebra 1 Self-Paced

Prealgebra 1
Tuesday, May 13 - Aug 26
Thursday, May 29 - Sep 11
Sunday, Jun 15 - Oct 12
Monday, Jun 30 - Oct 20
Wednesday, Jul 16 - Oct 29

Prealgebra 2 Self-Paced

Prealgebra 2
Wednesday, May 7 - Aug 20
Monday, Jun 2 - Sep 22
Sunday, Jun 29 - Oct 26
Friday, Jul 25 - Nov 21

Introduction to Algebra A Self-Paced

Introduction to Algebra A
Sunday, May 11 - Sep 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Wednesday, May 14 - Aug 27
Friday, May 30 - Sep 26
Monday, Jun 2 - Sep 22
Sunday, Jun 15 - Oct 12
Thursday, Jun 26 - Oct 9
Tuesday, Jul 15 - Oct 28

Introduction to Counting & Probability Self-Paced

Introduction to Counting & Probability
Thursday, May 15 - Jul 31
Sunday, Jun 1 - Aug 24
Thursday, Jun 12 - Aug 28
Wednesday, Jul 9 - Sep 24
Sunday, Jul 27 - Oct 19

Introduction to Number Theory
Friday, May 9 - Aug 1
Wednesday, May 21 - Aug 6
Monday, Jun 9 - Aug 25
Sunday, Jun 15 - Sep 14
Tuesday, Jul 15 - Sep 30

Introduction to Algebra B Self-Paced

Introduction to Algebra B
Tuesday, May 6 - Aug 19
Wednesday, Jun 4 - Sep 17
Sunday, Jun 22 - Oct 19
Friday, Jul 18 - Nov 14

Introduction to Geometry
Sunday, May 11 - Nov 9
Tuesday, May 20 - Oct 28
Monday, Jun 16 - Dec 8
Friday, Jun 20 - Jan 9
Sunday, Jun 29 - Jan 11
Monday, Jul 14 - Jan 19

Paradoxes and Infinity
Mon, Tue, Wed, & Thurs, Jul 14 - Jul 16 (meets every day of the week!)

Intermediate: Grades 8-12

Intermediate Algebra
Sunday, Jun 1 - Nov 23
Tuesday, Jun 10 - Nov 18
Wednesday, Jun 25 - Dec 10
Sunday, Jul 13 - Jan 18
Thursday, Jul 24 - Jan 22

Intermediate Counting & Probability
Wednesday, May 21 - Sep 17
Sunday, Jun 22 - Nov 2

Intermediate Number Theory
Sunday, Jun 1 - Aug 24
Wednesday, Jun 18 - Sep 3

Precalculus
Friday, May 16 - Oct 24
Sunday, Jun 1 - Nov 9
Monday, Jun 30 - Dec 8

Advanced: Grades 9-12

Olympiad Geometry
Tuesday, Jun 10 - Aug 26

Calculus
Tuesday, May 27 - Nov 11
Wednesday, Jun 25 - Dec 17

Group Theory
Thursday, Jun 12 - Sep 11

Contest Preparation: Grades 6-12

MATHCOUNTS/AMC 8 Basics
Friday, May 23 - Aug 15
Monday, Jun 2 - Aug 18
Thursday, Jun 12 - Aug 28
Sunday, Jun 22 - Sep 21
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

MATHCOUNTS/AMC 8 Advanced
Sunday, May 11 - Aug 10
Tuesday, May 27 - Aug 12
Wednesday, Jun 11 - Aug 27
Sunday, Jun 22 - Sep 21
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

AMC 10 Problem Series
Friday, May 9 - Aug 1
Sunday, Jun 1 - Aug 24
Thursday, Jun 12 - Aug 28
Tuesday, Jun 17 - Sep 2
Sunday, Jun 22 - Sep 21 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Monday, Jun 23 - Sep 15
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

AMC 10 Final Fives
Sunday, May 11 - Jun 8
Tuesday, May 27 - Jun 17
Monday, Jun 30 - Jul 21

AMC 12 Problem Series
Tuesday, May 27 - Aug 12
Thursday, Jun 12 - Aug 28
Sunday, Jun 22 - Sep 21
Wednesday, Aug 6 - Oct 22

AMC 12 Final Fives
Sunday, May 18 - Jun 15

AIME Problem Series A
Thursday, May 22 - Jul 31

AIME Problem Series B
Sunday, Jun 22 - Sep 21

F=ma Problem Series
Wednesday, Jun 11 - Aug 27

WOOT Programs
Visit the pages linked for full schedule details for each of these programs!


MathWOOT Level 1
MathWOOT Level 2
ChemWOOT
CodeWOOT
PhysicsWOOT

Programming

Introduction to Programming with Python
Thursday, May 22 - Aug 7
Sunday, Jun 15 - Sep 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Tuesday, Jun 17 - Sep 2
Monday, Jun 30 - Sep 22

Intermediate Programming with Python
Sunday, Jun 1 - Aug 24
Monday, Jun 30 - Sep 22

USACO Bronze Problem Series
Tuesday, May 13 - Jul 29
Sunday, Jun 22 - Sep 1

Physics

Introduction to Physics
Wednesday, May 21 - Aug 6
Sunday, Jun 15 - Sep 14
Monday, Jun 23 - Sep 15

Physics 1: Mechanics
Thursday, May 22 - Oct 30
Monday, Jun 23 - Dec 15

Relativity
Mon, Tue, Wed & Thurs, Jun 23 - Jun 26 (meets every day of the week!)
0 replies
jlacosta
May 1, 2025
0 replies
k i Adding contests to the Contest Collections
dcouchman   1
N Apr 5, 2023 by v_Enhance
Want to help AoPS remain a valuable Olympiad resource? Help us add contests to AoPS's Contest Collections.

Find instructions and a list of contests to add here: https://artofproblemsolving.com/community/c40244h1064480_contests_to_add
1 reply
dcouchman
Sep 9, 2019
v_Enhance
Apr 5, 2023
k i Zero tolerance
ZetaX   49
N May 4, 2019 by NoDealsHere
Source: Use your common sense! (enough is enough)
Some users don't want to learn, some other simply ignore advises.
But please follow the following guideline:


To make it short: ALWAYS USE YOUR COMMON SENSE IF POSTING!
If you don't have common sense, don't post.


More specifically:

For new threads:


a) Good, meaningful title:
The title has to say what the problem is about in best way possible.
If that title occured already, it's definitely bad. And contest names aren't good either.
That's in fact a requirement for being able to search old problems.

Examples:
Bad titles:
- "Hard"/"Medium"/"Easy" (if you find it so cool how hard/easy it is, tell it in the post and use a title that tells us the problem)
- "Number Theory" (hey guy, guess why this forum's named that way¿ and is it the only such problem on earth¿)
- "Fibonacci" (there are millions of Fibonacci problems out there, all posted and named the same...)
- "Chinese TST 2003" (does this say anything about the problem¿)
Good titles:
- "On divisors of a³+2b³+4c³-6abc"
- "Number of solutions to x²+y²=6z²"
- "Fibonacci numbers are never squares"


b) Use search function:
Before posting a "new" problem spend at least two, better five, minutes to look if this problem was posted before. If it was, don't repost it. If you have anything important to say on topic, post it in one of the older threads.
If the thread is locked cause of this, use search function.

Update (by Amir Hossein). The best way to search for two keywords in AoPS is to input
[code]+"first keyword" +"second keyword"[/code]
so that any post containing both strings "first word" and "second form".


c) Good problem statement:
Some recent really bad post was:
[quote]$lim_{n\to 1}^{+\infty}\frac{1}{n}-lnn$[/quote]
It contains no question and no answer.
If you do this, too, you are on the best way to get your thread deleted. Write everything clearly, define where your variables come from (and define the "natural" numbers if used). Additionally read your post at least twice before submitting. After you sent it, read it again and use the Edit-Button if necessary to correct errors.


For answers to already existing threads:


d) Of any interest and with content:
Don't post things that are more trivial than completely obvious. For example, if the question is to solve $x^{3}+y^{3}=z^{3}$, do not answer with "$x=y=z=0$ is a solution" only. Either you post any kind of proof or at least something unexpected (like "$x=1337, y=481, z=42$ is the smallest solution). Someone that does not see that $x=y=z=0$ is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
Similar, posting "I have solved this problem" but not posting anything else is not welcome; it even looks that you just want to show off what a genius you are.

e) Well written and checked answers:
Like c) for new threads, check your solutions at least twice for mistakes. And after sending, read it again and use the Edit-Button if necessary to correct errors.



To repeat it: ALWAYS USE YOUR COMMON SENSE IF POSTING!


Everything definitely out of range of common sense will be locked or deleted (exept for new users having less than about 42 posts, they are newbies and need/get some time to learn).

The above rules will be applied from next monday (5. march of 2007).
Feel free to discuss on this here.
49 replies
ZetaX
Feb 27, 2007
NoDealsHere
May 4, 2019
Self-evident inequality trick
Lukaluce   1
N 19 minutes ago by Sadigly
Source: 2025 Junior Macedonian Mathematical Olympiad P4
Let $x, y$, and $z$ be positive real numbers, such that $x^2 + y^2 + z^2 = 3$. Prove the inequality
\[\frac{x^3}{2 + x} + \frac{y^3}{2 + y} + \frac{z^3}{2 + z} \ge 1.\]When does the equality hold?
1 reply
Lukaluce
22 minutes ago
Sadigly
19 minutes ago
Intersections and concyclic points
Lukaluce   1
N 20 minutes ago by Ianis
Source: 2025 Junior Macedonian Mathematical Olympiad P2
Let $B_1$ be the foot of the altitude from the vertex $B$ in the acute-angled $\triangle ABC$. Let $D$ be the midpoint of side $AB$, and $O$ be the circumcentre of $\triangle ABC$. Line $B_1D$ meets line $CO$ at $E$. Prove that the points $B, C, B_1$, and $E$ lie on a circle.
1 reply
2 viewing
Lukaluce
27 minutes ago
Ianis
20 minutes ago
An algorithm for discovering prime numbers?
Lukaluce   0
24 minutes ago
Source: 2025 Junior Macedonian Mathematical Olympiad P3
Is there an infinite sequence of prime numbers $p_1, p_2, ..., p_n, ...,$ such that for every $i \in \mathbb{N}, p_{i + 1} \in \{2p_i - 1, 2p_i + 1\}$ is satisfied? Explain the answer.
0 replies
1 viewing
Lukaluce
24 minutes ago
0 replies
Difficult combinatorics problem
shactal   2
N 30 minutes ago by aaravdodhia
Can someone help me with this problem? Let $n\in \mathbb N^*$. We call a distribution the act of distributing the integers from $1$
to $n^2$ represented by tokens to players $A_1$ to $A_n$ so that they all have the same number of tokens in their urns.
We say that $A_i$ beats $A_j$ when, when $A_i$ and $A_j$ each draw a token from their urn, $A_i$ has a strictly greater chance of drawing a larger number than $A_j$. We then denote $A_i>A_j$. A distribution is said to be chicken-fox-viper when $A_1>A_2>\ldots>A_n>A_1$ What is $R(n)$
, the number of chicken-fox-viper distributions?
2 replies
shactal
5 hours ago
aaravdodhia
30 minutes ago
Batman chases the Joker on a square board
Lukaluce   0
32 minutes ago
Source: 2025 Junior Macedonian Mathematical Olympiad P1
Batman, Robin, and The Joker are in three of the vertex cells in a square $2025 \times 2025$ board, such that Batman and Robin are on the same diagonal (picture). In each round, first The Joker moves to an adjacent cell (having a common side), without exiting the board. Then in the same round Batman and Robin move to an adjacent cell. The Joker wins if he reaches the fourth "target" vertex cell (marked T). Batman and Robin win if they catch The Joker i.e. at least one of them is on the same cell as The Joker.

If in each move all three can see where the others moved, who has a winning strategy, The Joker, or Batman and Robin? Explain the answer.

Comment. Batman and Robin decide their common strategy at the beginning.

IMAGE
0 replies
Lukaluce
32 minutes ago
0 replies
Symmedian line
April   93
N 44 minutes ago by aidenkim119
Source: All Russian Olympiad - Problem 9.2, 10.2
Let be given a triangle $ ABC$ and its internal angle bisector $ BD$ $ (D\in BC)$. The line $ BD$ intersects the circumcircle $ \Omega$ of triangle $ ABC$ at $ B$ and $ E$. Circle $ \omega$ with diameter $ DE$ cuts $ \Omega$ again at $ F$. Prove that $ BF$ is the symmedian line of triangle $ ABC$.
93 replies
April
May 10, 2009
aidenkim119
44 minutes ago
Probably a good lemma
Zavyk09   1
N 44 minutes ago by Zavyk09
Source: found when solving exercises
Let $ABC$ be a triangle with circumcircle $\omega$. Arbitrary points $E, F$ on $AC, AB$ respectively. Circumcircle $\Omega$ of triangle $AEF$ intersects $\omega$ at $P \ne A$. $BE$ intersects $CF$ at $I$. $PI$ cuts $\Omega$ and $\omega$ at $K, L$ respectively. Construct parallelogram $KFRE$. Prove that $A, R, P$ are collinear.
1 reply
Zavyk09
3 hours ago
Zavyk09
44 minutes ago
Ez comb proposed by ME
IEatProblemsForBreakfast   1
N an hour ago by n1g3r14n
A and B play a game on two table:
1.At first one table got $n$ different coloured marbles on it and another one is empty
2.At each move player choose set of marbles that hadn't choose either players before and all chosen marbles from same table, and move all the marbles in that set to another table
3.Player who can not move lose
If A starts and they move alternatily who got the winning strategy?
1 reply
IEatProblemsForBreakfast
Today at 9:02 AM
n1g3r14n
an hour ago
geometry
luckvoltia.112   0
an hour ago
ChGiven an acute triangle ABC inscribed in circle $(O)$ The altitudes $BE, CF$ , intersect
each other at $H$. The tangents at $B$ and $C $of $(O)$ intersect at $S$. Let $M $be the midpoint of $BC$. $EM$ intersects $SC$
at $I$, $FM$ intersects $SB$ at $J.$
a) Prove that the points $I, S, M, J$ lie on the same circle.
b) The circle with diameter $AH$ intersects the circle $(O)$ at the second point $T.$ The line $AH$ intersects
$(O)$ at the second point $K$. Prove that $S,K,T$ are collinear.
0 replies
luckvoltia.112
an hour ago
0 replies
Hard Number Theory Problem
ZeltaQN2008   0
an hour ago
Source: VIMONI Test 2025
Let $n$ be a positive integer. Define $N_1$ as the number of integer pairs $(x,y)$ satisfying $x^{2}+3y^{2}=8n+4$ with $x$ odd. Define $N_2$ as the number of integer pairs $(x,y)$ satisfying $x^{2}+3y^{2}=8n+4.$
Prove that $N_{1}= \frac23\,N_{2}.$

0 replies
ZeltaQN2008
an hour ago
0 replies
Hard math inequality
noneofyou34   1
N an hour ago by JARP091
If a,b,c are positive real numbers, such that a+b+c=1. Prove that:
(b+c)(a+c)/(a+b)+ (b+a)(a+c)/(c+b)+(b+c)(a+b)/(a+c)>= Sqrt.(6(a(a+c)+b(a+b)+c(b+c)) +3
1 reply
noneofyou34
2 hours ago
JARP091
an hour ago
Exponents of integer question
Dheckob   4
N an hour ago by LeYohan
Find the smallest positive integer $m$ such that $5m$ is an exact 5th power, $6m$ is an exact 6th power, and $7m$ is an exact 7th power.
4 replies
Dheckob
Apr 12, 2017
LeYohan
an hour ago
Highest degree for 3-layer power tower (IMO ShortList 1991)
orl   36
N an hour ago by SomeonecoolLovesMaths
Source: IMO ShortList 1991, Problem 18 (BUL 1)
Find the highest degree $ k$ of $ 1991$ for which $ 1991^k$ divides the number \[ 1990^{1991^{1992}} + 1992^{1991^{1990}}.\]
36 replies
orl
Aug 15, 2008
SomeonecoolLovesMaths
an hour ago
ISI 2025
Zeroin   0
an hour ago
Let $\mathbb{N}$ denote the set of natural numbers and let $(a_i,b_i),1 \leq i \leq 9$ denote $9$ ordered pairs in $\mathbb{N} \times \mathbb{N}$. Prove that there exist $3$ distinct elements in the set $2^{a_i}3^{b_i}$ for $1 \leq i \leq 9$ whose product is a perfect cube.
0 replies
Zeroin
an hour ago
0 replies
Generating Functions
greenplanet2050   7
N Apr 30, 2025 by rchokler
So im learning generating functions and i dont really understand why $1+2x+3x^2+4x^3+5x^4+…=\dfrac{1}{(1-x)^2}$

can someone help

thank you :)
7 replies
greenplanet2050
Apr 29, 2025
rchokler
Apr 30, 2025
Generating Functions
G H J
G H BBookmark kLocked kLocked NReply
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
greenplanet2050
1324 posts
#1
Y by
So im learning generating functions and i dont really understand why $1+2x+3x^2+4x^3+5x^4+…=\dfrac{1}{(1-x)^2}$

can someone help

thank you :)
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Yiyj
26 posts
#2
Y by
Note that $1+2x+3x^2+\cdots$ is an arithmetico-geometric sequence. Then, we have the formula \[\sum_{k=1}^{\infty} x_k = \dfrac{dg_2}{(1-r)^2} + \dfrac{x_1}{1-r},\]where $d$ is the common difference of the arithmetic sequence, $r$ is the common ratio of the geometric sequence, $g_2$ is the second term of the geometric sequence, and $x_k$ are the terms of the arithmetico-geometric sequence.

Plugging in $d=1, r=x, g_2=x, x_1=1$, we get \[1+2x+3x^2+\cdots=\dfrac{x}{(1-x)^2}+\dfrac{1}{1-x} = \dfrac{x}{(1-x)^2}+\dfrac{1-x}{(1-x)^2} = \boxed{\dfrac{1}{(1-x)^2}}.\]Hope that helped!
This post has been edited 1 time. Last edited by Yiyj, Apr 29, 2025, 10:55 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Shan3t
382 posts
#3
Y by
greenplanet2050 wrote:
So im learning generating functions and i dont really understand why $1+2x+3x^2+4x^3+5x^4+…=\dfrac{1}{(1-x)^2}$

can someone help

thank you :)

Let $1+2x+3x^2\cdots = S.\quad(1)$

Now multiply $S,$ by $x,$ to get:

$x+2x^2+3x^3+4x^4+\cdots = S\cdot x\quad(2)$

Just subtract equation $2$ from equation $1,$ to get $1+x+x^2+x^3\cdots = \frac{1}{1-x} = S-S\cdot x.$ Simplify this, gives $S(1-x)=\frac1{1-x}\implies S=\frac{1}{(1-x)^2}.$
This post has been edited 1 time. Last edited by Shan3t, Apr 29, 2025, 11:02 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
martianrunner
207 posts
#4
Y by
Notice that $(1+x+x^2+x^3...)^2 = 1+2x+3x^2+4x^3...$ (this can be elementarily proven with induction by counting the pairs for each coefficient's term)

Since the value of a geometric sequence that goes $1+x+x^2+x^3...$ is $\frac{1}{1-x}$, we square that to get our answer of $\frac{1}{(1-x)^2}$
This post has been edited 2 times. Last edited by martianrunner, Apr 30, 2025, 2:36 AM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
greenplanet2050
1324 posts
#5
Y by
Thank you all!!
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Shan3t
382 posts
#6
Y by
greenplanet2050 wrote:
Thank you all!!

np :D

also @2bove sol very clean :D
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
ohiorizzler1434
786 posts
#7 • 1 Y
Y by compoly2010
It's a highly technical concept that combines convergence of geometric sequences with calculus, to represent the power series of a function around a point!
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
rchokler
2975 posts
#8
Y by
In general, the ring $\mathbb{C}[x]$ of all polynomials with complex coefficients is spanned by the basis $\{p_n\}_{n=0}^\infty$ where $p_n(x)=(x+1)(x+2)\cdots(x+n)$ under finite linear combinations. Note that $p_0(x)=1$ since it is the empty product.

Using this and the power rule for derivatives, we can find a formula for $\sum_{n=0}^\infty p(n)x^n$ for any polynomial $p$ and any $x\in(-1,1)$.

The idea is that $S_k(x)=\sum_{n=0}^\infty p_k(n)x^n=\sum_{n=0}^\infty\frac{d^k}{dx^k}x^{n+k}=\sum_{n=-k}^\infty\frac{d^k}{dx^k}x^{n+k}=\sum_{n=0}^\infty\frac{d^k}{dx^k}x^n=\sum_{n=0}^\infty\frac{d^k}{dx^k}x^n=\frac{d^k}{dx^k}\sum_{n=0}^\infty x^n=\frac{d^k}{dx^k}\frac{1}{1-x}=\frac{k!}{(1-x)^{k+1}}$.

So all you have to do is write $p$ of degree $n$, as $p=\sum_{k=0}^nc_kp_k$.

Example:
Find $\sum_{n=0}^\infty(n^3+5n^2-3n+4)x^n$.

Solution:
Use $p_0(n)=1$, $p_1(n)=n+1$, $p_2(n)=(n+1)(n+2)=n^2+3n+2$, and $p_3=(n+1)(n+2)(n+3)=n^3+6n^2+11n+6$.

$c_0p_0(n)+c_1p_1(n)+c_2p_2(n)+c_3p_3(n)=c_3n^3+(c_2+6c_3)n^2+(c_1+3c_2+11c_3)n+(c_0+c_1+2c_2+6c_3)\equiv n^3+5n^2-3n+4$
$\implies\begin{cases}c_3=1\\c_2+6c_3=5\\c_1+3c_2+11c_3=-3\\c_0+c_1+2c_2+6c_3=4\end{cases}\implies(c_0,c_1,c_2,c_3)=(11,-11,-1,1)$

Therefore $\sum_{n=0}^\infty(n^3+5n^2-3n+4)x^n=11S_0(x)-11S_1(x)-S_2(x)+S_3(x)=\frac{11}{1-x}-\frac{11}{(1-x)^2}-\frac{2}{(1-x)^3}+\frac{6}{(1-x)^4}=\frac{-11x^3+22x^2-9x+4}{(1-x)^4}$.
Z K Y
N Quick Reply
G
H
=
a