Summation and Series: Rules for Adding Sequences, Summaries of Law

Download Summation and Series: Rules for Adding Sequences and more Summaries Law in PDF only on Docsity! Sums & Series Suppose a 1 , a 2 , ... is a sequence. Sometimes we’ll want to sum the first k numbers (also known as terms) that appear in a sequence. A shorter way to write a 1 +a 2 +a 3 + · · ·+a k is as kX i=1 a i There are four rules that are important to know when using P . They are listed below. In all of the rules, a 1 , a 2 , a 3 , ... and b 1 , b 2 , b 3 , ... are sequences and c 2 R. Rule 1. c kX i=1 a i = kX i=1 ca i Rule #1 is the distributive law. It’s another way of writing the equation c(a 1 + a 2 + · · ·+ a k ) = ca 1 + ca 2 + · · ·+ ca k Rule 2. kX i=1 a i + kX i=1 b i = kX i=1 (a i + b i ) This rule is essentially another form of the commutative law for addition. It’s another way of writing that (a 1 +a 2 + · · ·+a k )+ (b 1 + b 2 + · · ·+ b k ) = (a 1 + b 1 )+ (a 2 + b 2 )+ · · ·+(a k + b k ) Rule 3. kX i=1 a i kX i=1 b i = kX i=1 (a i b i ) 25 Rule #3 is a combination of the first two rules. To see that, remember that b i = (1)b i , so we can use Rule #1 (with c = 1) followed by Rule #2 to derive Rule #3, as is shown below: kX i=1 a i kX i=1 b i = kX i=1 a i + kX i=1 b i = kX i=1 (a i + (b i )) = kX i=1 (a i b i ) Rule 4. kX i=1 c = kc The fourth rule can be a little tricky. The number c does not depend on i — it’s a constant — so P k i=1 c is taken to mean that you should add the first k terms in the sequence c, c, c, c, .... That is to say that kX i=1 c = c+ c+ · · ·+ c = kc Examples. • P 5 i=1 2 means that you should add the first 5 terms of the constant sequence 2, 2, 2, 2, 2, . . .. That is, 5X i=1 2 = 2 + 2 + 2 + 2 + 2 = 5(2) = 10 • P 20 i=1 3 = 20(3) = 60 * * * * * * * * * * * * * 26 Examples. • The sum of the terms in the sequence 1, 1 2 , 1 4 , 1 8 , ... equals 2. We know the sequence is geometric, follows the rule a n+1 = 1 2 a n , and that the first term in the sequence equals 1. Thus 1 + 1 2 + 1 4 + 1 8 + ... = 1 1 1 2 = 1 1 2 = 2 • The sum of the terms in the sequence 5, 5 3 , 5 9 , 5 27 , ... equals 5 1 1 3 = 5 2 3 = 15 2 Caution. If a 1 , a 2 , a 3 , ... isn’t geometric, or if it is but either r 1 or r  1, then 1X i=1 a i probably doesn’t make sense. * * * * * * * * * * * * * 29 Exercises 3i + 2 describes a sequence. When i = 1, we have 3(1) + 2 = 5. When i = 2, we have 3(2) + 2 = 8. When i = 3, we have 3(3) + 2 = 11. 3i + 2 is the formula for the sequence 5, 8, 11, 14, 17, . . . . The sum 4X i=1 (3i+ 2) is what you’d get by adding the first 4 terms of the sequence described by 3i+ 2. That is, 4X i=1 (3i+ 2) = 5 + 8 + 11 + 14 = 38 The next three problems involve summing terms of formulas that are de- scribed by the formulas 2i 1, i2 2, and i 3. Find the sums. 1.) 5X i=1 (2i 1) 2.) 4X i=1 (i2 2) 3.) 3X i=1 i 3 Find the following sums using Rule #4 from page 26. 4.) 50X i=1 3 5.) 100X i=1 49 6.) 78X i=1 (2) Just as we used 3i + 2 at the top of the page as a formula for describing a sequence, so too i is a formula for describing a sequence. The sequence described by i is a very simple arithmetic sequence. The first term is 1, the second term is 2, the third term is 3, and so on, so that the sequence is 1, 2, 3, 4, 5, 6, . . .. Use the formula on page 27 to find the sums below, the sums of the first 40, 100, and 900 terms of this arithmetic sequence. 7.) 40X i=1 i 8.) 100X i=1 i 9.) 900X i=1 i 30 10.) What is the sum of the first 701 terms of the sequence 5,1, 3, 7, ...? 11.) What is the sum of the first 53 terms of the sequence 140, 137, 134, 131, ...? 12.) What is the sum of the first 100 terms of the sequence 4, 9, 14, 19, ...? 13.) What is the sum of the first 80 terms of the sequence 53, 54, 55, 56, ...? Notice that 2 6 i is a formula for a geometric sequence. When i = 1, 2 6 i = 2 6 1 = 1 3 . When i = 2, 2 6 i = 2 6 2 = 1 18 . When i = 3, 2 6 i = 2 6 3 = 1 54 . The formula 2 6 i describes the geometric sequence 1 3 , 1 18 , 1 54 , . . .. It’s a geometric sequence whose fist term is 1 3 , and whose remaining terms are each found by multiplying the preceding term by 1 6 . That is, this a geometric sequence where a 1 = 1 3 and r = 1 6 . Because 1 6 is between 1 and 1, we have a formula (on page 28) that tells us how to find the geometric series asked for in #14 below. Find the given geometric series in #14-16. 14.) 1X i=1 2 6i 15.) 1X i=1 7 3i 16.) 1X i=1 10 2i The problems in #17-21 are asking you to find a geometric series. They are the same type of problem as those in #14-16, they just perhaps look a little di↵erent. Find the first term of the sequence (a 1 ), find the number that each term of the sequence is multiplied by to get the next term of the sequence (r), and then use the same formula that you used in #14-16, as long as r is a number between 1 and 1. 17.) Sum all of the terms of the geometric sequence 20, 5, 5 4 , 5 16 , .... 18.) Sum all of the terms of the geometric sequence 120, 90, 135 2 , 405 8 , .... 19.) Sum all of the terms of the geometric sequence 7, 14 3 , 28 9 , 56 27 , .... 20.) Sum all of the terms of the geometric sequence 25, 15, 9, 27 5 , .... 21.) Sum all of the terms of the geometric sequence 1,1 2 , 1 4 ,1 8 , .... 31