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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
X^ k
i=
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
X^ k
i=
a (^) i =
X^ k
i=
cai
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 + · · · + cak
Rule 2.
X^ k
i=
a (^) i +
X^ k
i=
b (^) i =
X^ k
i=
(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.
X^ k
i=
ai
X^ k
i=
b (^) i =
X^ k
i=
(a (^) i b (^) i )
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:
X^ k
i=
a (^) i
X^ k
i=
b (^) i =
X^ k
i=
a (^) i +
X^ k
i=
b (^) i
X^ k
i=
(a (^) i + ( b (^) i ))
X^ k
i=
(a (^) i b (^) i )
Rule 4.
X^ k
i=
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
X^ k
i=
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,
X^5
i=
P 20
i=1 3 = 20(3) = 60
a 1 = 1. Our formula a (^) n = a 1 +(n 1)d tells us that a 63 = 1+(62)3 = 185. Therefore, X^63
i=
a (^) i =
Example. The sum of the first 201 terms of the sequence 10, 17 , 24 , 31 , ... equals 2012 (10 + 1410) = 2012 (1420) = 142, 710.
Geometric series
It usually doesn’t make any sense at all to talk about adding infinitely many numbers. But if a 1 , a 2 , a 3 , ... is a geometric sequence where a (^) n+1 = ra (^) n and 1 < r < 1, 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 X^1
i=
ai
to represent adding all of the numbers in the sequence a 1 , a 2 , a 3 , ..., and we call this infinite “sum” a series. For the moment, let S = a 1 + a 2 + a 3 + a 4 + · · ·. Remember that in a geometric sequence a (^) n = r n ^1 a 1 , so we can rewrite S as
S = a 1 + ra 1 + r 2 a 1 + r 3 a 1 + · · ·
Using the distributive law we can multiply both sides of the line above by r:
rS = ra 1 + r 2 a 1 + r 3 a 1 + · · · Now we can subtract rS from S. If we did, the ra 1 terms in S and rS would cancel. So would the r 2 a 1 terms, the r 3 a 1 terms, etc. Thus, S rS = a 1. Since the distributive law tells us that S rS = S(1 r), we have S(1 r) = a 1 , or in other words, S = 1 a ^1 r. We have shown that
X^1
i=
a (^) i =
a (^1) 1 r
Examples.
- The sum of the terms in the sequence 1, 12 , 14 , 18 , ... equals 2. We know the sequence is geometric, follows the rule a (^) n+1 = 12 a (^) n , and that the first term in the sequence equals 1. Thus
1 +
1 2
- The sum of the terms in the sequence 5, 53 , 59 , 275 , ... equals 5 1 (^13)
2 3
Caution. If a 1 , a 2 , a 3 , ... isn’t geometric, or if it is but either r 1 or r 1, then (^1) X
i=
a (^) i
probably doesn’t make sense.
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 (^62) i is a formula for a geometric sequence. When i = 1, (^62) i = 2 61 =^
1
- When^ i^ = 2,^
2 6 i^ =^
2 62 =^
1
- When^ i^ = 3,^
2 6 i^ =^
2 63 =^
1
- The formula^
2 6 i describes the geometric sequence 13 , 181 , 541 ,.. .. It’s a geometric sequence whose fist term is 13 , and whose remaining terms are each found by multiplying the preceding term by 16. That is, this a geometric sequence where a 1 = 13 and r = 16. Because 16 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.
X^1
i=
6 i^
X^1
i=
3 i^
X^1
i=
2 i
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 , 54 , 165 , ....
18.) Sum all of the terms of the geometric sequence 120, 90 , 1352 , 4058 , ....
19.) Sum all of the terms of the geometric sequence 7, 143 , 289 , 5627 , ....
20.) Sum all of the terms of the geometric sequence 25, 15 , 9 , 275 , ....
21.) Sum all of the terms of the geometric sequence 1, 12 , 14 , 18 , ....
22.) If the sum of the first 3976 terms of the sequence a 1 , a 2 , a 3 , ... equals 114, then what is the sum of the first 3976 terms of the sequence 32 a 1 , 32 a 2 , 32 a 3 , ...?
23.) If the sum of the first 20 terms of the sequence a 1 , a 2 , a 3 ,... equals 7, and the sum of the first 20 terms of the sequence b 1 , b 2 , b 3 ,... equals 13, then what is the sum of the first 20 terms of the sequence (a 1 + b 1 ), (a 2 + b 2 ), (a 3 + b 3 ),.. .?
24.) 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 $20. How much will you spend on gas in the next 8 years?
25.) Suppose you are entertaining two di↵erent job o↵ers. Job A has a starting salary of $20,000 and assures you of a raise of $1,000 per year. Job B o↵ers you a starting salary of $23,000, with a yearly raise of $725. Which job will pay you more over the first ten years? How much more?
26.) 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?
