problems with (!) function

Sep 2009
21
0
Im not sure what ! is called but my question is how is

(x+1)! / x! = (x+1)

also is there a formula to solve

(x+1)^a where a equals a # (for example 18)
 
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Dec 2009
872
381
1111
Im not sure what ! is called but my question is how is

(x+1)! / x! = (x+1)
Dear Reminisce,

\(\displaystyle \frac{(x+1)!}{x!}=\frac{(x+1) \times (x) \times (x-1) \times ...... \times 2 \times 1}{(x) \times (x-1) ..... \times 2 \times 1} =x+1\)

Did you understand now?
 
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Jul 2010
456
138
Im not sure what ! is called but my question is how is

(x+1)! / x! = (x+1)

\(\displaystyle 5!=5 \cdot 4 \cdot 3 \cdot 2 \cdot 1\)
or
\(\displaystyle 5!=5 \cdot4 !\)
\(\displaystyle n!=n \cdot (n-1) \cdot (n-1) \cdot (n-2) \cdot \cdot \cdot (n-m)!\)
so u have...
\(\displaystyle \frac{(x+1)!}{x!}=\frac{(x+1) \cdot x!}{x!}=(x+1)\)
 

chisigma

MHF Hall of Honor
Mar 2009
2,162
994
near Piacenza (Italy)
Im not sure what ! is called but my question is how is

(x+1)! / x! = (x+1)

also is there a formula to solve

(x+1)^a where a equals a # (for example 18)
In general is x is a real number with x>-1 [so that it isn't neccessarly an integer...] the factorial function [or 'Pi function' according to Gauss interpretation...] is defined as a definite integral that I don't report because the details are a little complex. If You want more details about it see...

Gamma function - Wikipedia, the free encyclopedia

In particular the 'facrorial function' satisfies the relation...

\(\displaystyle x! = x\ (x-1)!\) (1)

Till now we are 'all right'... a little less clear for Me is the second part of Your question, regarding the expression \(\displaystyle (x+1)^{a}\)... could You supply more details please?...

Kind regards

\(\displaystyle \chi\) \(\displaystyle \sigma\)
 
Sep 2009
21
0
Thank you very much
as for the second part i was wondering if there is a formula to FOIL anything bigger then 3
example
(x+1)^18
because it would be a hassle to write (x+1) eighteen times and FOILing it like that
 
Jul 2010
456
138
u can use : (sorry i do'nt know how this LaTex work just jet)

_____1_____
___1_2_1___
__1_3_3_1__
_1_4_6_4_1_
...................

for let's say 10th... it would be... 1_10_45_120_210_252_210_120_45_10_1

and u'll have
\(\displaystyle (x+1)^{10}=x^{10}+10x^{9}+45x^{8}+120x^{7}+210x^{6}+252x^{5}+210x^{4}+120x^{3}+45x^{2}+10x^{1}+1\)


and so on.... but if u need to know let's sey n-th member of that u can use binomial template (or something... I'm bad with English... more or less)
 
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Sep 2009
21
0
I'm still a bit confused. Like how did you get the values 1_10_45_120....
from using the pyramid
 
Jul 2010
456
138
well... when u do pyramid ... see that at each side are ones... but numbers inside are sum od two numbers from above...
look at 4... firs is one ... second is (above 1+3 =4) 4 .... 3rd is (above 3+3 =6) 6 ... 4th is (above 3+1=4) 4 and 5th is 1
and so on...

or like this :

\(\displaystyle (a+b)^n = a^n + \binom {n}{1}a^{n-1} b + \binom {n}{2}a^{n-2}b^2 + \cdot \cdot \cdot + \binom{n}{n-1}a b^{n-1}+b^n\)

or if u need let's say (k+1)th member u'll use this one :

\(\displaystyle T_{(k+1)}=\binom {n}{k}a^{n-k}b^k \) that's one if u need (k+1)th member from begining ...

\(\displaystyle T_{(k+1)}=\binom {n}{k}a^{k}b^{n-k} \) that's one if u need (k+1)th member from end ...
 
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undefined

MHF Hall of Honor
Mar 2010
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Chicago
I'm still a bit confused. Like how did you get the values 1_10_45_120....
from using the pyramid
Pascal's triangle - Wikipedia, the free encyclopedia

Also for doing binomial coefficients you might find this useful

Combinations and Permutations Calculator

Choose No - No from dropdown menus, then for example n = 18 and r = 5 will give you the coefficient in front of the x^5 term of (x+1)^18 expanded. See

Binomial coefficient - Wikipedia, the free encyclopedia