* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download 11.4 – Arithmetic Series
Survey
Document related concepts
List of first-order theories wikipedia , lookup
History of Grandi's series wikipedia , lookup
Location arithmetic wikipedia , lookup
Large numbers wikipedia , lookup
Law of large numbers wikipedia , lookup
Non-standard calculus wikipedia , lookup
Mathematics of radio engineering wikipedia , lookup
Collatz conjecture wikipedia , lookup
Laws of Form wikipedia , lookup
Elementary mathematics wikipedia , lookup
Transcript
11.4 – Arithmetic Series How do I know if it is an arithmetic series? • A series is the expression for the sum of the terms of a sequence, not just “what is the next terms Ex: 6, 9, 12, 15, 18 . . . Ex: 6 + 9 + 12 + 15 + 18 This is a list of the numbers in the pattern an not a sum. It is a sequence. Note it goes on forever, so we say it is an infinite sequence. Here we are adding the values. We call this a series. Because it does not go on forever, we say it is a finite series. Note: if the numbers go on forever, it is infinite; if it has a definitive ending it is finite. Evaluating a Series • Simply add up the values. Ex: 2, 11, 20, 29, 38, 47 2+11+20+29+38+47 = 147 Easy Cheesy! But isn’t there a quicker way to do this??? Sum of a Finite Arithmetic Series n Sn ( a1 an ) 2 Where: Sn is the sum of all the terms n = number of terms a1 = first term an = last term From our last example: 2+11+20+29+38+47 = 147 Sn 6 ( 2 47) 147 2 Let’s try one: evaluate the series: 5, 9, 13,17,21,25,29 Sn 7 (5 29) 119 2 Summation When we don’t want to write out a whole bunch of numbers in the series, the summation symbol is used when writing a series. The limits are the greatest and least values of n. Upper Limit (greatest value of n) Summation symbol Explicit function for the sequence Lower Limit (least value of n) So, the way this works is plug in n=1 to the equation and continue through n=3. (5*1 + 1) + (5*2 +1) + (5*3 + 1) = 33 Writing a series in summation form • Ex: 102 + 104 + 106 + 108 + 110 + 112 n = 6 terms 1st term = 1 Rule: Hmmmm. . . . Rule = 100 + 2n Let’s Evaluate: Sn n ( a1 an ) 2 Yes, you can add manually. But let’s try using the shortcut: Sn 6 (102 112) 642 2 Let’s try some Find the number of terms, the first term and the last term. Then evaluate the series: Let’s try some Find the number of terms, the first term and the last term. Then evaluate the series: Why is the answer not 58? N = 10 1st = 1 Notice we can use the shortcut here: Last = 10 a1= 1-3 = -2 N=4 1st = 2 Last = 5 a10 = 10-3 = 7 4+9+16+25 = 54 Sn 10 ( 2 7) 5(5) 25 2 Note: this is NOT an arithmetic series. You can NOT use the shortcut; you have to manually crunch out all the values.