Sequences and series
Late in the eighteenth century, a five-year-old student and his German grammar school classmates were asked to find the sum of the first 100 counting numbers. Before the teacher had a chance to sit down, the five-year-old brought his slate to the teacher with the correct answer on it. The student, Carl F. Gauss (1777–1855), would grow up to be one of the world’s most famous mathematicians and physicists. His analysis of the problem was the beginning of the study of arithmetic sequences and series. In this chapter, you will learn more about these topics.
Summation notation
Gauss and his classmates were told to find the sum of the first 100 counting numbers. Before you learn how Gauss approached the problem, you will learn some notation for problems like this.
Restating the problem, find 1 + 2 + 3 + 4 + … + 97 + 98 + 99 + 100. Writing all 100 numbers would be very tedious. Mathematicians use the upper-case Greek letter sigma (Σ) to represent summation. The notation, 
rule, consists of three inputs: the rule that generates the numbers, the first input value, and the last input value. For example, 1 + 2 + 3 + 4 + … + 97 + 98 + 99 + 100 can be written more succinctly as 
Start with 1, then go to 2, then 3, and so on, until 100 is reached. Then, add the terms.
Expand each of the summation problems in questions 1–4.
1. 
2. 
3. 
4. 
Rewrite each of the following questions using summation notation.
5. 5 + 11 + 17 + 23 + 29 + 35
6. 120 + 115 + 110 + 105 + 100
7. 81 + 89 + 97 + 105 + … + 153
8. 2 + 4 + 8 + 16 + 32 + 64 + … + 2048
Recursion
A sequence is a listing of elements. The elements of the list are separated by commas.
Some sequences are more familiar than others.
S, M, T, W, T, F, S is the sequence for the first letters of the days of the week, beginning with Sunday.
1, 2, 3, 4, 5, … is a sequence of the counting numbers.
O, T, T, F, F, S, S, E, N, T is the sequence of the first letter of the first ten counting numbers when spelled out.
1, 1, 2, 3, 5, 8, 13, 21, … is the Fibonacci sequence. (This sequence is named for the medieval mathematician, Leonardo Fibonacci).
If you do not know about the Fibonacci sequence, it would be worth your while to do some research to learn more about Fibonacci—a mathematician of Italian birth who traveled and lived in northern Africa and the Mediterranean region during the twelfth and early thirteenth centuries—and the sequence named for him. Fibonacci promoted the Hindu–Arabic numeral system to Europeans.
1. Think of a number.
2. Add 5 to it.
3. Add 5 to the answer you just got.
4. Repeat step 3 until your answer is greater than 100.
How many terms did it take?
The answer for each step in the problem depends upon the number that came from the previous step. In essence, this is the recursive process. You start with a single term, or some fixed number of terms, and all subsequent terms depend upon the initial value(s).
Find the next five terms in the sequences given.
1. a1 = 7; an = an−1 + 9
2. a1 = 78; an = an−1 − 6
3. a1 = 4; an = 5 an−1
4. a1 = 2; an = 3 an−1 + 1
5. a1 = 6; a2 = 11; an = an−1 + an−2
6. a1 = 4; an = 8 (an−1)−2
7. 
Arithmetic sequences
An arithmetic sequence is one in which the difference between consecutive terms is a constant. Letting d represent the common difference, the nth term of an arithmetic sequence can be represented by the formula an = a1 + d(n − 1).
It should be noted that the equation for generating the nth term of an arithmetic sequence is a linear equation. an = a1 + d(n − 1) becomes an = a1 + dn − d or an = dn + a1 − d. Letting b = a1 − d, the equation becomes an = dn + b. The common difference is the slope of the line, and the difference between the first term and the common difference is the y-intercept of the line.
Determine the value of the specified term in the arithmetic sequences given.
1. The fortieth term of the arithmetic sequence where 914 is the first term and −7 is the common difference
2. The eightieth term of the arithmetic sequence where 87 is the first term and 19 is the common difference
3. The hundredth term of the sequence 8, 20, 32, 44, …
4. The seventy-fifth term of the sequence 1023, 1012, 1001, 1090, …
5. The fifty-seventh term of the arithmetic sequence in which a15 = 89 and a24 = 224
6. The hundred-twentieth term of the arithmetic sequence in which a5 = 9 and a34 = 212
Arithmetic series
In the story told at the beginning of this chapter, young Carl Gauss determined the sum of the first 100 counting numbers. The counting numbers form an arithmetic sequence and the sum of the numbers form an arithmetic series. (In general, a series is the sum of the terms of a sequence.) How was Gauss able to do the problem so easily? Gauss wrote S as the sum of the numbers, and he then rewrote the sum placing the numbers in reversed order.
Adding these equations to each other, he got
2S = 101 + 101 + 101 + … + 101 + 101 + 101.
The right-hand side of the equation has 101 added 100 times, so 2S = 100(101) or 
5050. Examining this process, you can see that 100 was the number of terms being added and that 101 is the sum of the first term (1) and the last term (100). The denominator, 2, comes from adding the two equations together. The formula for the sum of n terms of an arithmetic series is 
Solve the following.
1. Find the sum of the first 90 terms of an arithmetic series where 350 is the first term and −4 is the common difference.
2. Find the sum of the first 40 terms of an arithmetic series where −226 is the first term and 17 is the common difference.
3. Find S60 for the series 21 + 27 + 33 + 39 + ….
4. Find S225 for the series 7 + 11 + 15 + 19 + ….
5. Find the sum of the series 13 + 27 + 41 + 55 + … + 419.
6. Find the sum of the series 812 + 799 + 786 + 773 + … + 45.
Geometric sequences
A geometric sequence is one in which the ratio between consecutive terms is a constant. Letting r represent the common ratio, the nth term of a geometric sequence can be represented by the formula an = a1 × rn−1.
Note that the exponent is 1 in this case, not 0. This is because the phrase “one year after” represents the second term of the sequence, with the initial deposit being the first term.
Using what you have learned about geometric sequences, determine the information requested for each of the following.
1. Find the tenth term of a geometric sequence where 18 is the first term and 4 is the common ratio.
2. Find the twelfth term of a geometric sequence where 81 is the first term and 
is the common ratio.
3. Find the fifteenth term in the sequence 8, 12, 18, 27, . . . .
4. Find the ninth term in the sequence 80,000, 20,000, 5000, 2500, . . . .
5. Find the twentieth term of a geometric sequence in which the fifth term is 256 and the ninth term is 4096.
6. $10,000 is invested in an account that pays 2.5% compounded annually. How much money will be in the account after 15 years?
Geometric series
The sum of the terms in a geometric sequence is called a geometric series. The derivation for the formula for the sum of a geometric series is different from the way in which Gauss determined the sum of the first 100 counting numbers. Let Sn represent the sum of the first n terms of the geometric series.
Sn = a1 + a1r + a1r2 + a1r3 + … + a1rn−1
Multiply both sides of the equation by r, the common ratio.
rSn = a1r + a1r2 + a1r3 + … + a1rn−1 + arn
Notice how all but the first term in the first equation and the last term in the second equation have matches. Subtract the second equation from the first equation.
Sn − rSn = a1 − a1rn
Factor and solve for Sn to get 
.
Find the sums for the following series.
1. 
2. The first 20 terms in the series 20 + 60 + 180 + 540 + …
3. 120,000 + 60,000 + 30,000 + 15,000 + …
4. 0.72 + 0.0072 + 0.000072 + 0.00000072 + …
5. The first 15 terms of a geometric series if the fourth term is 24 and the ninth term is 768