Binomial Summation

Art of Problem Solving

AoPS Online

High School Math Grades 9-12, Ages 13-18

Grades 9-12, Ages 13-18

3 M G

BBookmark VNew Topic kLocked

High School Math Grades 9-12, Ages 13-18

Grades 9-12, Ages 13-18

3 M G

BBookmark VNew Topic kLocked

No tags match your search

M

Binomial Summation

G H J

Reveal topic tags

binomial theorem

L

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.

1299 posts

#1 h Yesterday at 1:43 AM

Y by

Given that: $l_n = \sum_{r=1}^{n-1} \frac{2r^2-r(n-2)+1}{(n-r) \binom{n}{r}}$ , find $l_{15}$ .

This post has been edited 1 time. Last edited by Saucepan_man02, Yesterday at 1:44 AM
Reason: EDIT

Z K Y

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

generatingFraction

2 posts

generatingFraction

#2 h Yesterday at 7:10 AM

Y by

It looks strange, how do you get this.

Z K Y

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

11 posts

#3 h Yesterday at 9:07 AM

Y by

$l_n = \sum_{r=1}^{n-1} \frac{2r^2-r(n-2)+1}{(n-r) \binom{n}{r}}$ $=\sum_{r=1}^{n-1}\frac{(r+1)^2-r(n-r)}{(n-r)\frac{n!}{r!(n-r)!}}$ $=\sum_{r=1}^{n-1}\frac{r!(n-r-1)![(r+1)^2-r(n-r)]}{n!}$ $=\frac{1}{n!}\sum_{r=1}^{n-1}r!(n-r-1)![(r+1)^2-r(n-r)]$ $=\frac{1}{n!}\sum_{r=1}^{n-1}[(r+1)(n-r-1)!(r+1)!-r(n-r)!r!]$ $=\frac{1}{n!}[n\cdot n!-(n-1)!]$ $=n-\frac{1}{n}$ $l_{15}=\frac{224}{15}$

Z K Y

N Quick Reply

G

H

=

© 2025 AoPS Incorporated

a