Lesson Arithmetic and Geometric Sequences and Series
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Algebra: Sequences of numbers, series and how to sum them
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A sequence is a set of numbers determined as either arithmetic, geometric, or neither. Examples: 1.) 1,2,3,4,5,6,7 are all seperated by + 1 ~> Arithmetic 2.) 1,3,9,27,81 are all seperated by * 3 ~> Geometric 3.) 1,4,9,16,25 are neither because they are not seperated by +, -, /, or * -------------------------------------------------------------------------------------------- * Arithmetic Sequence Equation ~> Tn = T0 + (n - 1)d Tn = term after "n" units T0 = first term n = the amount of numbers after the initial .... for example in (1.), 2 would be the second d = the common difference .... for example in (1.), all seperated by + 1 .... d = 1 ------------------------------------------------------------------------ Now, lets apply this: Sequence = 3, 7, 11, 15, 19, 23, 27, 31, 35, 39, 43, 47, 51 What is the 10th number? tn = t0 + (n - 1)d t0 = 3 n = 10 d = 4 tn = 3 + (9)4 = 3 + 36 = 39 What is the 13th number? tn = 3 + (12)4 = 3 + 48 = 51 -------------------------------------------------------------------------------------------- * Geometric Sequence Equation ~> Tn = t0 * r^(n - 1) t0 = first term r = common ratio .... for example in (2.), r = 3 .... way to find out ~> divide second number by first or divide third number by second or divide fourth number by third n = the nth number in the sequence ------------------------------------------------------------------------ Now, lets apply this: Sequence = 2, 4, 8, 16, 32, 64, 128, 256 What is the 6th number? t0 = 2 r = 2 n = 6 Tn = 2*2^(5) = 2^6 = 64 What is the 8th number? Tn = 2*2^(7) = 2^8 = 256 -------------------------------------------------------------------------------------------- * Arithmetic Series Equation ~> Sum = n(t0 + tn)/2 n = number of digits to sum ~> if you are trying to find the sum of 3 numbers ~> n = 3 t0 = first number tn = last number Different Version ~> Sum = n(t0 + t0 + (n - 1)d)/2 Sum = n(2*t0 + (n - 1)d)/2 ------------------------------------------------------------------------ Now, we should try to apply: Sequence = 0, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60 .... What is the sum of the first 4 digits? t0 = 0 n = 4 d = 6 Sum = n(2*t0 + (n - 1)d)/2 Sum = 4*(3)*6/2 = 2*3*6 = 36 What is the sum of the first 8 digits? Sum = n(2*t0 + (n - 1)d)/2 Sum = 8*(7)*6/2 = 4*7*6 = 168 -------------------------------------------------------------------------------------------- * Geometric Series The equation can be quite "breath-taking", but it is my favorite ... :) Sum = t0 + t0 * r + t0 * r^2 + t0 * r^3 + t0 * r^4 .... + t0 * r^(n - 1) ... reasoning ~> t0 (first term) + t0 * r (second term) .... Sum * r = t0 * r + t0 * r^2 + t0 * r^3 + t0 * r^4 .... + t0 * r^(n - 1) + t0 * r^(n) Sum * r - Sum = t0 * r^n - t0 Sum(r - 1) = t0(r^n - 1) Sum = t0*(r^n - 1)/(r - 1) = t0*(1 - r^n)/(1 - r) ------------------------------------------------------------------------ Now, we should try to apply: Sequence = 2, 6, 18, 54, 162, 486 .... What is the sum of the first 4 digits? t0 = 2 r = ? 6/2 = 3 18/6 = 3 r = 3 n = 4 Sum = t0*(1 - r^n)/(1 - r) Sum = 2*(1 - 3^4)/(1 - 3) Sum = 2*(1 - 81)/(-2) = -(1 - 81) = 80 What is the sum of the first 6 digits? Sum = t0*(1 - r^n)/(1 - r) Sum = 2*(1 - 3^6)/(1 - 3) Sum = 2*(1 - 729)/(-2) = -(1 - 729) = 728