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--- urp:algebra [2021-10-30]
nerf_herder
+++ urp:algebra [2022-02-01] (current)
nerf_herder [Arithmetic Sequences]
@@ Line 1: / Line 1: @@
 ====Algebra & Pre-Calc====
+Related pages:
   * [[Graphing|Graphing circles, ellipses, parabolas, hyperbolas]]
   * [[poly|Polynomials, quadratic formula and completing the square]]
+  * [[power|Powers, radicals (roots) & logs]]
-**quadratic formula**: x = (-b +- √(b²-4ac)) / 2a
+**quadratic formula**: x = (-b ±√(b²-4ac)) / 2a
-=== System of equations===
+**PEMDAS/BODMAS ** - order of operations: Parentheses/Brackets, Exponents/Order, Multiply-Divide, Add-Subtract
+==== System of equations====
 equations, for 2 unknowns (need n equations to solve for n unknowns)
@@ Line 25: / Line 28: @@
 x     = 19, solve for x, then put in to find y
-=== Parametric equations ===
+==== Parametric equations ====
 Define x and y in terms of t (time). Arrows on graph represent increasing values of t. Allows you to create functions, using two graphs, from things are not functions for both x and y together (circles, ellipses, x = y², etc)
@@ Line 33: / Line 36: @@
 https://courses.lumenlearning.com/suny-osalgebratrig/chapter/parametric-equations/
-=== Powers & Radicals ===
+==== Arithmetic/Geometric Sequences ====
+**Arithmetic sequence** has a constant difference between the terms, such as 1, 3, 5, 7, 9...
+Basic form to find term n: a(n) = a(1) + d(n-1), where d = step size (difference between terms), a(1) is the first term
+Sum of an arithmetic sequence:
+  S = n/2(a + L)
+    S = sum, n = # of terms, a = value of first term, L = value of last term
+  If don't know last term, just substitute a(n):
+    S = n/2 (2a + (n − 1) d)
+**Geometric sequence** terms are found by multiplying the previous term by a constant, such as 2, 4, 8, 16 ...
+a(n) = arⁿ⁻¹
+Other sequences exist:
+  * squares: a(n) = n², cubes, etc.
+  * triangular numbers: a(n) = n(n+1)/2 (number of dots in a triangle of n rows)
+  * fibonacci sequence: a(n) = a(n-1) + a(n-2)
+Convergence of a power sequence: http://math.bu.edu/people/prakashb/Teaching/32LS10/Lectures/11-2.pdf
+====Factorial====
+  n! = n * (n-1)!
+! = 1
+!/(7-3)! = 7!/4! = 7*6*5 * (4!/4!) = 7*6*5
+==== Functions ====
+Definitions:
+  * Function: only has one y value for any x value. Discontinuities are okay (breaks in allowed x values)
+  * One-to-one function: a function with only one x value for any y value.
+  * Domain: what values of x are described
+  * Range: resulting values of y coming from the function
+composition of functions: (f’g)(x) = f(g(x)), order of evaluation is important
+**Inverse** of a function - only possible if no two values of x produce the same result, ie. must be one-to-one (or limit the domain to make it so)
+  Example: To find f⁻¹(x) for f(x) = 5x + 3
+     y = 5x + 3
+     y-3 = 5x
+     x = (y-3)/5
+     f⁻¹(x) = (x-3)/5   (replace y with x on the last step,
+                         since x is input to the function, and y is output)
+Graphing an inverse: reflection of the graph about the line y=x
+**asymptotes**
+y = (quadratic1 of x) / (quadratic2 of x)
+  * vertical asymptotes (VA) are when denominator goes to zero
+  * horizontal asymptotes (HA) is when x goes to infinity, look at highest order of x in numerator and denominator:
+    *  y = axⁿ / bxᵏ
+    *  if n < k : HA = 0
+    *  if n = k : HA = a/b
+    *  if n > k : no HA
+      * if n = k+1 : oblique (diagonal) asymptote - approaches the line y = mx+b (from polynomial long division)
+      * if n > k+1 : no asymptote
+look halfway down the page here: https://www.mathsisfun.com/algebra/rational-expression.html
+====Interest, half life, amortization====
+Annual rate, continuous rate of growth:
+  Y = a*bᵗ
+  a = principle amount, b = annual growth, t = time (years)
+  y = a*eᵏᵗ = a*(eᵏ)ᵗ, and k = continuous rate of growth
+     k = ln(b)
+Half life:
+  Nt = No(1/2)^(t/t.5)
+   Nt = amount at time t, No = initial amount, t.5 = half-life time
+This can be rearranged to:
+  t.5 = t/(log0.5(Nt/No)) = t / (log(Nt/No)/log(1/2))
+Also,
+  Nt = No*e^(-t/tau)
+  tau = mean lifetime
+  tau = 1/lambda,  lambda = decay constant
+  t.5 = ln(2)/lambda
+  (ln(2) is close to 7 => rule of 70 for doubling returns??)
+====misc====
+abs. value of imaginary number
+  |a + bi| = √(a^2 + b^2)
-=== Logs ===
+__rational numbers__ can be expressed as a fraction of two integers. The decimal expansion either terminates or repeats.
-=== Functions ===
+__irrational__ includes square roots, pi, etc
 ==== See also ====

The Ungerecht Family Tree

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