Sivu:Lukion taulukot.pdf/20

MERKINTÖJÄ, MÄÄRITELMIÄ JA KAAVOJA 19

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4.	$a^{n}-b^{n}=\left(a-b\right)\left(a^{n-1}+a^{n-2}b+\ldots +ab^{n-2}+b^{n-1}\right)$ $a^{n}+b^{n}=\left(a+b\right)\left(a^{n-1}-a^{n-2}b+\ldots -ab^{n-2}+b^{n-1}\right)$, kun n on pariton luonnollinen luku	samankorkuisten [potenssin] erotus ja summa
5.	$P\left(x\right)=ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$, missä $P\left(x_{1}\right)=0,\ P\left(x_{2}\right)=0$

Kertomat

$n$	$n!$		$n$	$n!$
1	1		11	39 916 800
2	2		12	479 001 600
3	6		13	6 227 020 800
4	24		14	87 178 291 200
5	120		15	1 307 674 368 000
6	720		16	$\approx 2{,}09\cdot 10^{13}$
7	5 040		17	$\approx 3{,}56\cdot 10^{14}$
8	40 320		18	$\approx 6{,}40\cdot 10^{15}$
9	362 880		19	$\approx 1{,}22\cdot 10^{17}$
10	3 628 800		20	$\approx 2{,}43\cdot 10^{18}$

$n!=1\cdot 2\cdot \ldots \cdot \left(n-1\right)n$,
$0!=1$, $1!=1$

Stirlingin kaava

$n!\approx n^{n}\cdot e^{-n}\cdot {\sqrt {2\pi n}}$

Binomikertoimet

${\binom {n}{k}}={\frac {n\left(n-1\right)\left(n-2\right)\ldots \left(n-\left(k-1\right)\right)}{k!}}={\frac {n!}{k!\left(n-k\right)!}}$

$n$	${\binom {n}{0}}$	${\binom {n}{1}}$	${\binom {n}{2}}$	${\binom {n}{3}}$	${\binom {n}{4}}$	${\binom {n}{5}}$	${\binom {n}{6}}$	${\binom {n}{7}}$	${\binom {n}{8}}$
1	1	1
2	1	2	1
3	1	3	3	1
4	1	4	6	4	1
5	1	5	10	10	5	1
6	1	6	15	20	15	6	1
7	1	7	21	35	35	21	7	1
8	1	8	28	56	70	56	28	8	1

Newtonin binomikaava

${\begin{aligned}(a+b)^{n}&=a^{n}+{\binom {n}{1}}a^{n-1}b+{\binom {n}{2}}a^{n-2}b^{2}+\ldots +{\binom {n}{k}}a^{n-k}b^{k}+\ldots +b^{n}\\&=\sum _{k=0}^{n}{\binom {n}{k}}a^{n-k}b^{k}\end{aligned}}$