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urp:algebra
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| Both sides previous revision Previous revision Next revision | Previous revision | ||
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urp:algebra [2021-12-19] nerf_herder [Functions] |
urp:algebra [2022-02-01] nerf_herder [Arithmetic Sequences] |
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| https://courses.lumenlearning.com/suny-osalgebratrig/chapter/parametric-equations/ | https://courses.lumenlearning.com/suny-osalgebratrig/chapter/parametric-equations/ | ||
| - | ==== Arithmetic Sequences ==== | + | ==== 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 | 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 | ||
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| If don't know last term, just substitute a(n): | If don't know last term, just substitute a(n): | ||
| S = n/2 (2a + (n − 1) d) | 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 | Convergence of a power sequence: http://math.bu.edu/people/prakashb/Teaching/32LS10/Lectures/11-2.pdf | ||
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| * horizontal asymptotes (HA) is when x goes to infinity, look at highest order of x in numerator and denominator: | * horizontal asymptotes (HA) is when x goes to infinity, look at highest order of x in numerator and denominator: | ||
| * y = axⁿ / bxᵏ | * y = axⁿ / bxᵏ | ||
| - | * if n > k : no HA | ||
| * if n < k : HA = 0 | * if n < k : HA = 0 | ||
| * if n = k : HA = a/b | * if n = k : HA = a/b | ||
| - | * oblique (diagonal) - approaches the line y = mx+b | + | * if n > k : no HA |
| - | * look halfway down the page here: https://www.mathsisfun.com/algebra/rational-expression.html | + | * if n = k+1 : oblique (diagonal) asymptote - approaches the line y = mx+b (from polynomial long division) |
| - | * if power of x in numerator > power of x in denominator: asymptote = 0 | + | * if n > k+1 : no asymptote |
| - | * if power of x in numerator = power of x in denominator: asymptote = horizontal, not zero (ratio of largest coefficients) | + | |
| - | * if power of x in numerator = 1 less than power of x in denominator: asymptote = oblique (definition of line, from polynomial long division) | + | look halfway down the page here: https://www.mathsisfun.com/algebra/rational-expression.html |
| - | * if power of x in numerator > 1 less than power of x in denominator: asymptote = none | + | |
| ====Interest, half life, amortization==== | ====Interest, half life, amortization==== | ||
urp/algebra.txt · Last modified: 2022-02-01 by nerf_herder
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