Using sequence notation

One big clue that you’re dealing with a sequence is when you see something like or . The braces, , indicate that you have a list of items, called terms; commas usually separate the terms in a list from one another. The term is the notation for the rule that represents a particular sequence. When identifying a sequence, you can list the terms in the sequence, showing enough terms to establish a pattern, or you can give the rule that creates the terms.

For instance, if you see the notation , you know that the sequence consists of the terms . is the rule that creates the sequence when you insert all the positive integers in place of the n. When , ; when , ; and so on. The domain of a sequence is all positive integers (counting numbers), so the process is easy as 1, 2, 3.

Because the terms in a sequence are linked to positive integers, you can refer to them by their positions in the listing of the integers. If the rule for a sequence is , for example, the terms in the sequence are named , and so on. This nice, orderly arrangement allows you to ask for the tenth term in the sequence by writing . You don’t have to write the first nine terms to get to the tenth one. Sequence notation is a timesaver!

No-fear factorials in sequences

A mathematical operation you see in many sequences is the factorial. The symbol for factorial is an exclamation mark.

Here’s the formula for the factorial in a sequence: . When you perform a factorial, you multiply the number you’re operating on times every positive integer smaller than that number.

For example, , and .

You apply a special rule for 0! (zero factorial). The rule is that . “What? How can that possibly be?” Well, it is. Mathematicians discovered that assigning the number 1 to 0! makes every operation involving factorials work out better. (Why you need this rule for 0! becomes more apparent in Chapter 17. For now, just be content to use the rule when writing the terms in a sequence.)

So, if , you write that , , , and so on. You write the terms in the sequence as .

Alternating sequential patterns

One special type of sequence is an alternating sequence. An alternating sequence has terms that forever alternate back and forth from positive to negative to positive. The terms in an alternating sequence have a multiplier of , which is raised to some power such as n, , or . Adding the power, which is related to the number of the term, to the causes the terms to alternate because the positive integers alternate between even and odd. Even powers of are equal to , and odd powers of are equal to .

For instance, the alternating sequence , because

and so on.

Here’s an example of an alternating sequence that has a factorial (see the previous section) and fraction — a little bit of everything. The first four terms of the sequence are as follows:

So, the sequence is .

You can see how the absolute value of the terms keeps getting larger while the terms alternate between positive and negative (see Chapter 2 for more on absolute value).

Looking for sequential patterns

The list of terms in a sequence may or may not display an apparent pattern. Of course, if you see the function rule — the rule that tells you how to create all the terms in the sequence — you have a huge hint about the pattern of the terms. You can always list the terms of a sequence if you have the rule, and you can often create the rule when you have enough terms in the sequence to figure out the pattern.

The patterns you can look for range from simple to a bit tricky:

• A single-number difference between each term, such as 4, 9, 14, 19, … , where the difference between each term is 5
• A multiplier separating the terms, such as multiplying by 5 to get 2, 10, 50, 250, …
• A pattern within a pattern, such as with the numbers 2, 5, 9, 14, 20, … , where the differences between the numbers get bigger by one each time

When you have to figure out a pattern and write a rule for a sequence of numbers, you can refer to your list of possibilities — the ones I mention and others — and see which type of rule applies.

Difference between terms

The quickest, easiest pattern to find features a common difference between the terms. A difference between two numbers is the result of subtracting the previous term from the term in question. You can usually tell when you have a sequence of this type by inspecting it — looking at how far apart the numbers sit on the number line.

When looking for differences in a list of numbers, be careful that you always subtract in the same order — the number minus the number just to the left of it.

The following three sequences have something in common: The terms in the sequences have a common first difference, a common second difference, or a common third difference.

FIRST DIFFERENCE

When the first difference of the terms in a sequence is a constant number, the rule determining the terms is usually a linear expression (a linear expression has an exponent of 1 on n; see Chapter 2). For example, the sequence of numbers consists of terms that have a common difference of 5.

The rule for this example sequence is . You use the multiplier 5 to make the terms in the sequence each 5 more than the previous term. You subtract the 3 because when you replace n with 1, you get a number too big; you want to start with the number 2, so you subtract 3 from the first multiple. Sequences with a common first difference are called arithmetic sequences (I cover these sequences thoroughly in the section “Taking Note of Arithmetic and Geometric Sequences”).

SECOND DIFFERENCE

When the second difference of the terms in a sequence is a constant, like 2, the rule for that sequence is usually quadratic (see Chapter 3) — it contains the term . The sequence of numbers , for example, consists of terms that have a common second difference of 2. The first differences between the terms increase by two for each interval:

The rule used to create this example sequence is .

I can’t give you a quick, easy way to find the specific rules, but if you know that a rule should be quadratic, you have a place to start. You can try squaring the numbers 1, 2, 3, and so on, and then see how you have to “adjust” the squares by subtracting or adding so that the numbers in the sequence appear according to the rule.

THIRD DIFFERENCE

The sequence features a common third difference of 6. In the following diagram, you see that the row below the sequences shows the first differences; under the first differences are the second differences; and, finally, the third row contains the third differences:

The rule for this example sequence is , which contains a cubed term.

The rule for this sequence doesn’t necessarily leap off the page when you look at the terms. You have to play around with the terms a bit to determine the rule. Start with a cubed term and then try subtracting or adding constant numbers. If that doesn’t help, try adding or subtracting squares of numbers or just multiples of numbers. Sounds sort of haphazard, but you have limited options without the use of some calculus. Graphing calculators have curve-fitting features that take data and figure out the rules for you, but you still have to choose which kind of rules (what powers) make the data work.

Multiples and powers

Some sequences have fairly apparent rules that generate their terms, because each term is a multiple or a power of some constant number. For instance, the sequence consists of multiples of 3, and its rule is {3n}.

But what if the sequence starts with 21? What’s the rule for ? The terms are all multiples of 3, but {3n} doesn’t work because you have to start with . Remember, the domain or input of a sequence is made up of positive integers , so you can’t use anything smaller than 1 for n.

The way to get around starting sequences with smaller numbers is to add a constant to n (which is like a counter). The number 21 is , so add 6 to to form the rule .

The sequence has two interesting features: The terms alternate signs (see the section “Alternating sequential patterns”), and the fractions have consecutive integers in their denominators. To write the rule for this sequence, consider the two features. The alternating terms suggest a multiplier of . The first, third, fifth, and all other odd terms are positive, so you can create alternating multiples of that are positive by raising factor to . This makes those exponents even when n is odd and all the odd terms positive. For the fractions, you can put n, the term’s number, in the denominator. The rule for this sequence, therefore, is as follows:

Other sequences can have terms that are all powers of the same number. These sequences are called geometric sequences (I discuss them at great length in the section “Taking Note of Arithmetic and Geometric Sequences”). An example of a geometric sequence is . You can see that these terms are powers of the number 2, and the rule for the terms is .

Finding common ground: Arithmetic sequences

Arithmetic sequences (pronounced air-ith-mat-ick, with the emphasis on mat) are sequences whose terms have the same differences between them, no matter how far down the lists you go (in other words, how many terms the lists include).

One way to describe the general formula for arithmetic sequences is with the following:

The formula says that the nth term of the sequence is equal to the term directly before it (the term) plus the common difference, d.

Another equation you can use with arithmetic sequences is the following:

This formula says that the nth term of the sequence is equal to the first term, , plus times the common difference, d.

The equation you use depends on what you want to accomplish. You use the first formula if you single out a term in the sequence and want to find the next one in line. For example, if you want the next term after the number 201 in the sequence , you just add 3 to the 201 and get that the next term is 204. You use the second formula if you want to find a specific term in a sequence. For example, if you want the 50th term in the sequence where , you replace the n with 50, subtract the 1, multiply by 7, add 5, and you get 348. All that may sound like a lot of work, but it’s easier than listing all 50 terms. And you can solve for one of these rules for a sequence if you have just the right information.

For instance, if you know that the common difference between the terms of an arithmetic sequence is 4 and that the sixth term is 37, you can substitute this information into the equation , letting , , and . You get the following: , which becomes . Subtracting 20 from each side of the equation, you get that . Now, armed with the first term and the difference, you can write the general term using . Replacing with 17 and d with 4, you have . Distributing the 4 and simplifying, you have .

An arithmetic sequence that has a common difference of 4 and whose sixth term is 37 has the general rule .

You use this procedure when you’re given the number values of the terms flat out and when you have to figure them out from some application or story problem. Here’s an example of a story problem for which you need to use an arithmetic sequence.

You and a group of friends have been hired to be ushers at a local theatre performance, and your payment includes free tickets to the show. The theatre allocates the whole last row in the middle section for your group. The first row in the middle section of the theatre has 26 seats, and the number of seats in each row increases by one seat per row as you move backward for a total of 25 rows. How many seats are in the last row?

You can solve this problem quickly with an arithmetic sequence. Using the formula , you replace with 26, n with 25, and d with 1:

The last row has 50 seats. It looks like your friends can bring their friends, too!

Taking the multiplicative approach: Geometric sequences

A geometric sequence is a sequence in which each term is different from the one that follows it by a common ratio. In other words, the sequence has a constant number that multiplies each term to create the next one. With arithmetic sequences, you add a constant; with geometric sequences, you multiply the constant.

A general formula or rule for a geometric sequence is as follows:

In this equation, r is the constant ratio that multiplies each term. The rule says that, to get the nth term, you multiply the term before it — the term — by the ratio, r.

Another way you can write the general rule for a geometric sequence is as follows:

The second form of the rule involves the first term, , and applies the ratio as many times as needed. The nth term is equal to the first term multiplied by the ratio times.

You use the first rule, , when you’re given the ratio, r, and a particular term in the sequence, and you want the next term. If the ninth term in a sequence whose ratio is 3 is 65,610, for example, and you want the tenth term, you multiply 65,610 times 3 to get 196,830. You use the second rule, , when you’re given the first term in the sequence and you want a particular term. For instance, if you know that the first term is 3, the ratio is 2, and you want the 10th term, you find that term by multiplying 3 times , which is 1,536.

You can always find the ratio or multiplier, r, if you have two consecutive terms in a geometric sequence. Just divide the second term by the term immediately preceding it — the quotient is the ratio. For example, if you have a geometric sequence where the sixth term is 1,288,408 and the fifth term is 117,128, you can find r by dividing 1,288,408 by 117,128 to get 11. The ratio, r, is 11.

Say that the rule for a particular geometric sequence is . When , the power on the fraction is 0, and you have 360 multiplying the number 1. So, . When , the exponent is equal to 1, so the fraction multiplies the 360, and you get 120. Here are the first few terms in this sequence:

You can find each term by multiplying the previous term by .

Here’s another example to familiarize you with geometric formulas … feel free to break it out at parties in the future. An unwise gambler bets a dollar on the flip of a coin and loses. Instead of paying up, he says, “Double or nothing,” meaning that he wants to flip the coin again; he’ll pay two bucks if he loses, and his opponent gets nothing if the gambler wins. Oops! He loses again, and again he says, “Double or nothing!” If he repeats this doubling-and-losing process 20 times, how much will he owe on the 21st try?

Using the formula (you know the first term — the one dollar — and the multiplier), you replace the first term, , with the number 1, r with 2 for the doubling, and n with 21:

The gambler will owe over one million dollars if he keeps going to 21. If he’s unwilling to part with his first dollar, how’s he going to deal with this number?

Maybe the gambler shouldn’t start with such a big bet at the beginning. What if he starts with a quarter rather than a dollar? Using the same formula, , the first term is 0.25, and the ratio is still 2. So, . Looks like he’s still in big trouble. And, in two more flips, letting , he reaches the same amount lost in 21 flips with the dollar bill.

Introducing summation notation

Mathematicians like to keep formulas and rules neat and concise, so they created a special symbol to indicate that you’re adding the terms of a sequence. The special notation is sigma, , or summation notation.

The notation indicates that you want to add all the terms in the sequence that have the general rule , all the way from through :

For instance, if you want the sum , you need to find the first five terms, letting , and so on; you finish by adding all those terms:

You find that the sum is 45.

If you want to add all the terms in a sequence, forever and ever, you use the symbol in the sigma notation, which looks like .

Summing arithmetically

An arithmetic sequence has a general rule (see the section “Finding common ground: Arithmetic sequences”) that involves the first term and the common difference between consecutive terms: . An arithmetic series is the sum of the terms that come from an arithmetic sequence. Consider the arithmetic sequence . The first ten terms in this sequence are 4, 9, 14, 19, 24, 29, 34, 39, 44, and 49. The sum of these ten terms is 265. How did I get that value? Pencil and paper, my friend. Simple addition works just fine for a small list of numbers. However, your algebra teacher may not always bless you with small lists. Consider, now, a formula for the sum of the first n terms of an arithmetic sequence.

The sum of the first n terms of an arithmetic sequence, , is

Here, and d are the first term and difference, respectively, of the arithmetic sequence . The n indicates which term in the sequence you get when you put the value of n in the formula.

The first part of the formula, on the left, allows you to insert the first term, the difference, and the number of terms you want to add. The second part, on the right, is quicker and easier; you use it when you know both the first and last terms and how many terms you want to add. In the next example, n is 10, because that’s how many terms you’re adding. Inserting the 10 for n in the formula for the general term, you get 49.

To use the formula for the sum of the ten numbers 4, 9, 14, 19, 24, 29, 34, 39, 44, and 49 (previously added to get 265), you insert the known data: .

Now, say you want to add the first 100 numbers in a sequence that starts with 13 and features a common difference of 2 between each of the terms: , all the way to the 100th number. You find the sum of these 100 numbers by using the first part of the sum formula:

Summing geometrically

A geometric sequence consists of terms that differ from one another by a common ratio. You multiply a term in the sequence by a constant number or ratio to find the next term. You can use two different formulas to find the sum of the terms in a geometric sequence. You use the first formula to find the sum of a certain, finite number of terms of a geometric sequence — any geometric sequence at all. The second formula applies only to geometric sequences that have a ratio whose absolute value lies between zero and one (a proper fraction); you use it when you want to add all the terms in the sequence — forever and ever (for more on geometric sequences, see the section “Taking the multiplicative approach: Geometric sequences”).

Adding the first n terms

The formula you use to add a specific, finite number of terms from a geometric sequence involves a fraction where you subtract the ratio — or a power of the ratio — from one. You can’t reduce the formula, so don’t try. Just use it as it is.

You find the sum of the first n terms of the geometric sequence with the following formula:

The term is the first term of the sequence, and r represents the common ratio.

For example, say you want to add the first ten terms of the geometric sequence . You identify the first term, the 1, and then the ratio by which you multiply, 3. Substitute this information into the formula:

Quite a big number! Isn’t using the formula easier than adding ?

Adding all the terms to infinity

Geometric sequences have a ratio, or multiplier, that changes one term into the next one in line. If you multiply a number by 4 and the result by 4 and keep going, you create huge numbers in a short amount of time. So, it may sound impossible to add numbers that seem to get infinitely large.

But algebra has a really wonderful property for geometric sequences with ratios between negative one and one. The absolute values of the numbers in these sequences get smaller and smaller, and the sums of the terms in these sequences never exceed set, constant values.

If the ratio is bigger than one, the sum just grows and grows, and you get no final answer. If the ratio is negative and between 0 and , the sum approaches a single, constant value. For ratios smaller than , you have chaos again.

In Figure 16-1, you see the terms in a sequence that starts with 1 and has a ratio of . You also see the sum of all the terms up to and including that term.

The general rule for this particular sequence is as follows:

So, you’re really finding powers of . Here are the first few terms:

As you see in Figure 16-1, the sum of the terms is equal to almost two as the number of terms increases. The number in the numerator of the fraction of the sum is always one less than twice the denominator. The sum in Figure 16-1 approaches two but never exactly hits it. The sum gets so very, very close, though, that you can round up to two. This business of approaching a particular value is true of any geometric sequence with a proper fraction (between zero and one) for a ratio.

Algebra also offers a formula for finding the sum of all the terms in a geometric sequence with a ratio between zero and one. You may think this formula should be much more complicated than the formula for finding only a few terms, but that isn’t the case. This formula is actually much simpler.

The sum of all the terms from a geometric sequence whose ratio, r, has an absolute value between zero and one is , where is the first term in the sequence.

You can apply this rule to the sum of the sequence whose first term is 1 and whose common ratio is . .

You may be wondering what happens when the ratio is a negative number. Do the positive and negative terms cancel one another out? Not so. First, using the formula, I find the sum of all the terms of the geometric sequence whose first term is 1 and ratio is : . The sum is smaller than the sequence with a positive ratio, but the positives seem to outweigh the negatives!

Stacking the blocks

You are teaching your young nephew to stack blocks and show him how to create a flat pyramid where each row has one more block in it than the one preceding it. So the first row has one block, the second has two blocks, and so on. The local toy store is having a sale on block sets, and the biggest set has 210 blocks in it. How many rows of blocks will your nephew have in his pyramid if he uses all the blocks?

You can solve this problem by using the formula for the sum of the terms in an arithmetic sequence. The first row will have 1 block, and the difference in the number of blocks from row to row is 1. Using , you replace the sum, with 210, the first term, with 1, and the difference, d, with 1. Then solve for n, the number of rows.

Multiply each side of the equation by 2, and you have . Distribute the n, and then write the equation as a quadratic set equal to 0. You have . Factoring, . The solutions are and 20, but only the 20 works for a number of blocks. Your nephew will have a stack of blocks that has 20 rows!

Negotiating your allowance

You approach your dad about increasing your allowance because $10 per week just doesn’t cut it anymore. He replies, “Absolutely not. Not until you improve your math grade.” You then negotiate a deal with the following proposition: You’ll take 1 cent on the first day of the month, 2 cents on the second day of the month, 4 cents on the third, 8 cents on the fourth, and so on, doubling the amount each day until the end of the month. At that point, your dad can check on your math grade and see if he wants to change the system back and raise your allowance. He knows how crafty you are, so he asks you to explain what you’re up to before he agrees.

How much will you get with your system for the month of January? That month has 31 days of penny-pinching allowance. Using the formula for the sum of a geometric sequence whose first term is 1, common ratio is 2, and number of terms is 31 (see the section “Summing geometrically”), you calculate the following:

Of course, your answer is in pennies, so you move the decimal point over two places. Your allowance comes to a total of $21,474,836.47. What does your dad think of your math ability now? Unfortunately, he wants you to care about your grades as much as your money. No deal.

Bouncing a ball

You want to figure the total distance (up and down and up and down …) that a superball travels in n bounces, plus the first drop, if it always bounces back 75 percent of the distance it falls. You drop the superball onto a smooth sidewalk from a window sitting 40 feet off the ground. Look at Figure 16-2 to see what the problem entails.

As you can see, the first distance is the 40-foot drop. The ball bounces up 75 percent of the distance it fell, or 30 feet, and drops 30 feet again. It then bounces back 75 percent of 30 feet, or 22.5 feet, and repeats the process. Except for the initial 40 feet, all the measures double to account for going upward and then downward.

The question is: How far does the superball travel in 10 bounces, plus that first drop? To figure this out by using a geometric sequence (see the section “Summing geometrically”), you let the first term equal 30, the ratio equal 0.75, and the number of terms equal 10. You double that sum of the sequence to account for the distance the ball travels both up and down, and you add the first 40 feet to get the total distance traveled by the ball.

First, you find the value of the sum of the sequence:

Now you double the sum of the sequence and add 40 feet:

The ball travels over 266 feet in 10 bounces. Well done!

The formula you use to find the total distance traveled by a bouncing ball is built from the formula for the sum of a geometric series. In general, if the starting distance above the ground is h, and the percentage that the object bounces back each time is p, then you use these values in the formula, doubling the sum and adding on h. The first term in the series is ph, which is the percentage times the initial height.

Just fill in the initial height, h, and the percentage, p, as a decimal.

So, if you decide to climb the 897 steps to the top of the Washington Monument and drop a superball that bounces back 60 percent of the dropped height, then how far will the ball travel before stopping, if you’re able to drop it out a window 550 feet off the ground?

Using the formula I just created, you have

The ball travels a total of 2,200 feet.