Sums & Series. a i. i=1

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1 Sums & Series Suppose a,a,... 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 + a + a a k is as There are four rules that are important to know when using P. They are listed below. In all of the rules, a,a,a 3,... and b,b,b 3,... are sequences and c R. Rule. c = c Rule # is the distributive law. It s another way of writing the equation c(a + a + + a k )=ca + ca + + ca k Rule. + b i = ( + b i ) This rule is essentially another form of the commutative law for addition. It s another way of writing that (a + a + + a k )+(b + b + + b k )=(a + b )+(a + b )+ +(a k + b k ) Rule 3. b i = ( b i ) 5

2 Rule #3 is a combination of the first two rules. To see that, remember that b i =( )b i, so we can use Rule # (with c = ) followed by Rule # to derive Rule #3, as is shown below: b i = = = + b i ( +( b i )) ( b i ) Rule 4. c = kc The fourth rule can be a little tricky. The number c does not depend on i it saconstant so P k 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 c = c + c + + c = kc Examples. P 5 meansthatyoushouldaddthefirst5termsoftheconstant sequence,,,,,... That is, 5X =++++=5()=0 P 0 3=0(3)=60 * * * * * * * * * * * * * 6

3 Sum of first k terms in an arithmetic sequence If a,a,a 3,...is an arithmetic sequence, then a n+ = a n +d for some d R. We want to show that = k (a + a n ) To show this, let s write the sum in question in two di erent ways: front-toback, and back-to-front. That is, = a +(a + d)+(a +d)+ +(a k d)+(a k d)+a k and = a k +(a k d)+(a k d)+ +(a +d)+(a + d)+a Add the two equations above top-to-bottom to get =[a + a k ]+[a + a k ]+[a + a k ]+ +[a + a k ]+[a + a k ]+[a + a k ] Count and check that there are exactly k of the [a + a k ] terms in the line above being added. Thus, = k[a + a k ] which is equivalent to what we were trying to show: = k (a + a k ) Example. What is the sum of the first 63 terms of the sequence,, 5, 8,...? The sequence above is arithmetic, because each term in the sequence is 3 plus the term before it, so d =3. Thefirsttermofthesequenceis, so 7

4 a =. Our formula a n = a +(n )d tells us that a 63 = +(6)3 = 85. Therefore, X63 = ( +85)= (84) = 5, 796 Example. The sum of the first 0 terms of the sequence 0, 7, 4, 3,... equals 0 0 (0 + 40) = (40) = 4, 70. * * * * * * * * * * * * * Geometric series It usually doesn t make any sense at all to talk about adding infinitely many numbers. But if a,a,a 3,... is a geometric sequence where a n+ = ra n and < r <, then we can make sense of adding all of the terms of the sequence together. (We ll give some reason why this is in the chapter Geometric Series, after we ve looked at exponential functions.) We will use the symbols to represent adding all of the numbers in the sequence a,a,a 3,..., and we call this infinite sum a series. For the moment, let S = a + a + a 3 + a 4 +. Remember that in a geometric sequence a n = r n a, so we can rewrite S as S = a + ra + r a + r 3 a + Using the distributive law we can multiply both sides of the line above by r: rs = ra + r a + r 3 a + Now we can subtract rs from S. If we did, the ra terms in S and rs would cancel. So would the r a terms, the r 3 a terms, etc. Thus, S rs = a. Since the distributive law tells us that S rs = S( r), we have S( r) =a, or in other words, S = a r. We have shown that = a r 8

5 Examples. The sum of the terms in the sequence,, 4, 8,... equals. We know the sequence is geometric, follows the rule a n+ = a n, and that the first term in the sequence equals. Thus = = = The sum of the terms in the sequence 5, 5 3, 5 9, 5 7,... equals 5 3 = 5 3 = 5 Caution. If a,a,a 3,...isn t geometric, or if it is but either r orr apple, then probably doesn t make sense. * * * * * * * * * * * * * 9

6 Exercises 3i + describes a sequence. When i =, we have 3() + = 5. When i =, we have 3() + = 8. When i =3,wehave3(3)+=. 3i +is the formula for the sequence 5, 8,, 4, 7,... The sum 4X (3i +) is what you d get by adding the first 4 terms of the sequence described by 3i +. Thatis, 4X (3i +)=5+8++4=38 The next three problems involve summing terms of formulas that are described by the formulas i, i, and i 3. Find the sums..) 5X (i ).) 4X (i ) 3.) 3X i 3 Find the following sums using Rule #4 from page 6. 4.) X ) X ) X78 ( ) Just as we used 3i +atthetopofthepageasaformulafordescribing 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, the second term is, the third term is 3, and so on, so that the sequence is,, 3, 4, 5, 6,... Use the formula on page 7 to find the sums below, the sums of the first 40, 00, and 900 terms of this arithmetic sequence. 7.) X40 i 8.) X00 i 9.) X900 i 30

7 0.) What is the sum of the first 70 terms of the sequence 5,, 3, 7,...?.) What is the sum of the first 53 terms of the sequence 40, 37, 34, 3,...?.) What is the sum of the first 00 terms of the sequence 4, 9, 4, 9,...? 3.) What is the sum of the first 80 terms of the sequence 53, 54, 55, 56,...? Notice that 6 is a formula for a geometric sequence. When i =, i 6 = i 6 = 3. When i =, 6 = i 6 = 8. When i =3, 6 = i 6 = The formula 6 i describes the geometric sequence 3, 8, 54,... It s a geometric sequence whose fist term is 3, and whose remaining terms are each found by multiplying the preceding term by 6. That is, this a geometric sequence where a = 3 and r = 6. Because 6 is between and,wehaveaformula(onpage8)that tells us how to find the geometric series asked for in #4 below. Find the given geometric series in # ) 6 i 5.) 7 3 i 6.) 0 i The problems in #7- are asking you to find a geometric series. They are the same type of problem as those in #4-6, they just perhaps look a little di erent. Find the first term of the sequence (a ), 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 #4-6, as long as r is anumberbetween and. 7.) Sum all of the terms of the geometric sequence 0, 5, 5 4, 5 6,... 8.) Sum all of the terms of the geometric sequence 0, 90, 35, 405 8,... 9.) Sum all of the terms of the geometric sequence 7, 4 3, 8 9, 56 7,... 0.) Sum all of the terms of the geometric sequence 5, 5, 9, 7 5,....) Sum all of the terms of the geometric sequence,, 4, 8,... 3

8 .) If the sum of the first 3976 terms of the sequence a,a,a 3,... equals 4, then what is the sum of the first 3976 terms of the sequence 3 a, 3 a, 3 a 3,...? 3.) If the sum of the first 0 terms of the sequence a,a,a 3,... equals 7, and the sum of the first 0 terms of the sequence b,b,b 3,... equals 3, then what is the sum of the first 0 terms of the sequence (a + b ), (a + b ), (a 3 + b 3 ),...? 4.) Suppose that you expect to pay $400 for gas for your car next year, and that each year after that you plan your yearly gas expenditures will increase by $0. How much will you spend on gas in the next 8 years? 5.) Suppose you are entertaining two di erent job o ers. Job A has a starting salary of $0,000 and assures you of a raise of $,000 per year. Job Bo ers you a starting salary of $3,000, with a yearly raise of $75. Which job will pay you more over the first ten years? How much more? 6.) An oil well currently produces 5 million gallons of oil per year, but the well is drying up, and each year it will produce 60% of what it did the year before. How much oil can be produced from the well before it is completely dry? 3