# problems with (!) function

#### Reminisce

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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#### Sudharaka

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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#### yeKciM

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...
$$\displaystyle \frac{(x+1)!}{x!}=\frac{(x+1) \cdot x!}{x!}=(x+1)$$

#### Mathelogician

Infact it's called Factorial.

#### chisigma

MHF Hall of Honor
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

$$\displaystyle \chi$$ $$\displaystyle \sigma$$

#### Reminisce

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

#### yeKciM

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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#### Reminisce

I'm still a bit confused. Like how did you get the values 1_10_45_120....
from using the pyramid

#### yeKciM

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