CHAPTER 19 KEY FORMULAS AND EQUATIONS

Arithmetic sequences	Recursion formula	$a_{n} = a_{n - 1} + d$ $a_{n} = a_{n - 1} + d$ (19.1)
	`n`th term	$a_{n} = a_{1} + (n - 1) d$ $a_{n} = a_{1} + (n - 1) d$ (19.2)
	Sum of `n` terms	$S_{n} = \frac{n}{2} (a_{1} + a_{n})$ $S_{n} = \frac{n}{2} (a_{1} + a_{n})$ (19.3)
Geometric sequences	Recursion formula	$a_{n} = r a_{n - 1}$ $a_{n} = r a_{n - 1}$ (19.4)
	`n`th term	$a_{n} = a_{1} r^{n - 1}$ $a_{n} = a_{1} r^{n - 1}$ (19.5)
	Sum of `n` terms	$S_{n} = \frac{a_{1} (1 - r^{n})}{1 - r} (r \neq 1)$ $S_{n} = \frac{a_{1} (1 - r^{n})}{1 - r} (r \neq 1)$ (19.6)
Sum of an infinite geometric series	$S = \frac{a_{1}}{1 - r} (\| r \| < 1)$ $S = \frac{a_{1}}{1 - r} (\| r \| < 1)$ (19.7)
Factorial notation	$n! = n (n - 1) (n - 2) \dots (2) (1)$ $n! = n (n - 1) (n - 2) \dots (2) (1)$ (19.8)
Binomial formula	${(a + b)}^{n} = a^{n} + n a^{n - 1} b + \frac{n (n - 1)}{2!} a^{n - 2} b^{2} + \frac{n (n - 1) (n - 2)}{3!} a^{n - 3} b^{3} +^{} \dots + b^{n}$ ${(a + b)}^{n} = a^{n} + n a^{n - 1} b + \frac{n (n - 1)}{2!} a^{n - 2} b^{2} + \frac{n (n - 1) (n - 2)}{3!} a^{n - 3} b^{3} +^{} \dots + b^{n}$ (19.9)
Binomial series	${(1 + x)}^{n} = 1 + n x + \frac{n (n - 1)}{2!} x^{2} + \frac{n (n - 1) (n - 2)}{3!} x^{3} + \dots$ ${(1 + x)}^{n} = 1 + n x + \frac{n (n - 1)}{2!} x^{2} + \frac{n (n - 1) (n - 2)}{3!} x^{3} + \dots$ (19.10)

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Table of Contents for CHAPTER 19 KEY FORMULAS AND EQUATIONS