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Chapter 3

Sigma Notation and Series

Consider the sequence 10.20, 11.40, 12.10, 13.40 where each term represents the amount of money you earned as interest on your savings account for each of four years.

The sum of the terms, 10.2 + 11.4 + 12.1 + 13.4, represents the total interest you earned in the four year period. Such a sequence summation is called a series and is designated by Sn where n

represents the number of terms of the sequence being added.

Sn is often called an nth partial sum, since it can represent the sum of a certain "part" of a sequence.

A series can be represented in a compact form, called summation notation, or sigma notation.

The Greek capital letter sigma, , is used to indicate a sum.

"The summation from 1 to 4 of 3n":

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Examples:

Problem: Answer:

1. Evaluate:

2. Evaluate:

Notice how only the variable i was replaced with the values

from 2 to 4.

3. Evaluate:

Notice how raising (-1) to a power affected the signs of the terms. This is an important pattern strategy to remember.

4. Evaluate:

While the starting value is usually 1, it can actually be any integer value.

5. Use sigma notation to represent 2 + 4 + 6 + 8 + ... for 45 terms.

Look for a pattern based upon the position of each term. In this problem, each term is its position location times 2, for a sequence formula of

term position term

1 2 2 4 3 6 n 2n

One possible answer:

6. Use sigma notation to represent -3 + 6 - 9 + 12 - 15 + ... for 50 terms.

Again, look for a pattern. Each term is its position location times 3, but with signs alternating. Example #3 showed how to create alternating signs using powers of -1.

term position term

1 -3 2 6 3 -9 4 12

One possible answer:

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Strategies to remember when trying to find an expression for a sequence (series):

Example Possible notation (partial sum) Strategy

Look to see if a value is being consistently added (or

subtracted)

OR

Be aware that there is more than one answer.

Patterns can increase or decrease.

Look to see if a value is being consistently multiplied (or

divided)

Look to see if the values are "famous" numbers such as

perfect squares.

Look to see if the signs alternate. Alternating signs can be handled using powers of -1.

Practice with Sigma Notation and Series

Answer the following questions pertaining to sigma notation.

1.

Find S4 for the sequence 2, 4, 6, 8, 10, 12, 14, 16, 18, 20.

Choose:

12

20

110

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2.

Find S5 for the sequence:

Choose:

-22

10

22

3.

Evaluate:

Choose:

9

10

12

4.

Evaluate:

Choose:

7

9

11

5.

What is the difference between the

sum of the series and the sum of the series

?

Choose:

3

12

no difference, they are the same.

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6.

Evaluate:

Choose:

not possible

10

10x

7.

Use sigma notation to represent

3 + 6 + 9 + 12 + ...

for 28 terms.

Choose:

8.

Use sigma notation to represent

-3 + 6 - 12 + 24 - 48 + ...

for 35 terms.

Choose:

9.

Use sigma notation to represent:

for n terms.

Choose:

both work

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Problems for further investigation

1. Determine if the following statement is true or false for n = 8.

2. Evaluate where x1 = 5, x2 = 6, x3 = 3, x4 = 9

3. Find where x1 = 120 lbs, x2 = 135 lbs, x3 = 115 lbs

and x4 = 150 lbs. W is the mean weight of four teens.

4. True or False???

5. True or False???

Sequences

A sequence is an ordered list of numbers. The sum of the terms of a sequence is called a series.

• Each number of a sequence is called a term (or element) of the sequence.

• A finite sequence contains a finite number of terms (you can count them). 1, 4, 7, 10, 13

• An infinite sequence contains an infinite number of terms (you cannot count them). 1, 4, 7, 10, 13, . . .

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• The terms of a sequence are referred to in the subscripted form shown below,

where the natural number subscript refers to the location (position) of the term in the sequence.

(If you study computer programming languages such as C, C++, and

Java, you will find that the first position in their arrays (sequences) start

with a subscript of zero.)

• The general form of a sequence is represented:

• The domain of a sequence consists of the counting numbers 1, 2, 3, 4, ...

and the range consists of the terms of the sequence.

• The terms in a sequence may, or may not, have a pattern, or a related formula.

For some sequences, the terms are simply random.

Let's examine some sequences that have patterns:

Sequences often possess a definite patterarrive at the sequence's term

It is often possible to express such patternsthe sequence shown at the left, an explicit

where n represents the term's position in

 

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Practice with Sequences

Answer the following questions pertaining to sequences.

1.

Find the third term of the sequence

Choose:

3

6

9

2.

Write the first three terms of the sequence:

Choose:

0, 3, 8

3, 8, 15

1, 2, 3

3.

Find the 12th term of the sequence

Choose:

146

168

196

4.

Find the 8th term of the sequence

Choose:

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5.

Write a formula for the sequence 4, 8, 12, 16, 20, ...

Choose:

6.

Find the 15th term of the sequence

Choose:

196

-225

225

7.

Find the 12th term of the sequence

Choose:

-144

144

-e144

e144

Arithmetic Sequences and Series A sequence is an ordered list of numbers. The sum of the terms of a sequence is called a series.

While some sequences are simply random values, other sequences have a definite pattern that is used to arrive at the sequence's terms.

Two such sequences are the arithmetic and geometric sequences. Let's investigate the arithmetic sequence.

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Arithmetic Sequences ADD

If a sequence of values follows a pattern of adding a fixed amount from one term to the

next, it is referred to as an arithmetic sequence. The number added to each term is

constant (always the same).

The fixed amount is called the common difference, d, referring to the fact that the difference between two successive terms

yields the constant value that was added. To find the common difference, subtract the first

term from the second term.

Notice the linear nature of the scatter plot of the terms of an arithmetic sequence. The domain consists of the counting numbers 1, 2, 3, 4, ... and the range consists of the terms of the

sequence. While the x value increases by a constant value of one, the y value increases by a constant value of 3 (for this graph).

Examples:

Arithmetic Sequence Common Difference, d

1, 4, 7, 10, 13, 16, ... d = 3 add 3 to each term to arrive at the next term, or...the difference a2 - a1 is 3.

15, 10, 5, 0, -5, -10, ... d = -5 add -5 to each term to arrive at the next term, or...the difference a2 - a1 is -5.

add -1/2 to each term to arrive at the next term, or....the difference a2 - a1 is -1/2.

Formulas used with arithmetic sequences and arithmetic series:

To find any term of an arithmetic sequence:

where a1 is the first term of the sequence,

d is the common difference, n is the number of the term to find.

To find the sum of a certain number of terms of an arithmetic sequence:

where Sn is the sum of n terms (nth partial sum),

a1 is the first term, an is the nth term.

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Note: a1 is often simply referred to as a.

Examples:

Question Answer 1. Find the common difference for this arithmetic sequence 5, 9, 13, 17 ...

1. The common difference, d, can be found by subtracting the first term from the second term, which in this problem yields 4. Checking shows that 4 is the difference between all of the entries.

2. Find the common difference for the arithmetic sequence whose formula is an = 6n + 3

2. The formula indicates that 6 is the value being added (with increasing multiples) as the terms increase. A listing of the terms will also show what is happening in the sequence (start with n = 1). 9, 15, 21, 27, 33, ... The list shows the common difference to be 6.

3. Find the 10th term of the sequence 3, 5, 7, 9, ...

3. n = 10; a1 = 3, d = 2

The tenth term is 21.

4. Find a7 for an arithmetic sequence where a1 = 3x and d = -x.

4. n = 7; a1 = 3x, d = -x

5. Find t15 for an arithmetic sequence where t3 = -4 + 5i and t6 = -13 + 11i

5. Notice the change of labeling from a to t. The letter used in labeling is of no importance. Get a visual image of this problem

Using the third term as the "first" term, find the common difference from these known terms.

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Now, from t3 to t15 is 13 terms. t15 = -4 + 5i + (13-1)(-3 +2i) = -4 + 5i -36 +24i = -40 + 29i

6. Find a formula for the sequence 1, 3, 5, 7, ...

6. A formula will relate the subscript number of each term to the actual value of the term.

Substituting n = 1, gives 1. Substituting n = 2, gives 3, and so on.

7. Find the 25th term of the sequence -7, -4, -1, 2, ...

7. n = 25; a1 = -7, d = 3

8. Find the sum of the first 12 positive even integers.

8. The word "sum" indicates the need for the sum formula. positive even integers: 2, 4, 6, 8, ... n = 12; a1 = 2, d = 2 We are missing a12, for the sum formula, so we use the "any term" formula to find it.

Now, let's find the sum:

9. Insert 3 arithmetic means between 7 and 23.

Note: An arithmetic mean is the term between any two terms of an arithmetic sequence. It is simply the average (mean) of the given

9. While there are several solution methods, we will use our arithmetic sequence formulas. Draw a picture to better understand the situation. 7, ____, ____, ____, 23 This set of terms will be an arithmetic sequence. We know the first term, a1, the last term, an, but not the common difference, d. This question

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terms.

makes NO mention of "sum", so avoid that formula. Find the common difference:

Now, insert the terms using d. 7, 11, 15, 19, 23

10. Find the number of terms in the sequence 7, 10, 13, ..., 55.

10. a1 = 7, an = 55, d = 3. We need to find n. This question makes NO mention of "sum", so avoid that formula.

When solving for n, be sure your answer is a positive integer. There is no such thing as a fractional number of terms in a sequence!

11. A theater has 60 seats in the first row, 68 seats in the second row, 76 seats in the third row, and so on in the same increasing pattern. If the theater has 20 rows of seats, how many seats are in the theater?

11. The seating pattern is forming an arithmetic sequence. 60, 68, 76, ... We wish to find "the sum" of all of the seats. n = 20, a1 = 60, d = 8 and we need a20 for the sum.

Now, use the sum formula:

There are 2720 seats.

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Geometric Sequences and Series

A sequence is an ordered list of numbers.

The sum of the terms of a sequence is called a series.

While some sequences are simply random values, other sequences have a definite pattern that is used to arrive at the sequence's terms.

Two such sequences are the arithmetic and geometric sequences. Let's investigate the geometric sequence.

Geometric Sequences

MULTIPLY

If a sequence of values follows a pattern of multiplying a fixed amount (not zero) times each term to arrive at the following term, it is referred to as a geometric sequence. The

number multiplied each time is constant (always the same).

The fixed amount multiplied is called the common ratio, r, referring to the fact that the ratio (fraction) of the second term to the first term yields this common multiple. To find

the common ratio, divide the second term by the first term.

Notice the non-linear nature of the scatter plot of the terms of a geometric sequence. The domain consists of the counting numbers 1, 2, 3, 4, ... and the range consists of the terms of the

sequence. While the x value increases by a constant value of one, the y value increases by multiples of two (for this graph).

Examples:

Geometric Sequence Common Ratio, r

5, 10, 20, 40, ... r = 2 multiply each term by 2 to arrive at the next term or...divide a2 by a1 to find the common ratio, 2.

-11, 22, -44, 88, ... r = -2 multiply each term by -2 to arrive at the next term .or...divide a2 by a1 to find the common ratio, -2.

multiply each term by 2/3 to arrive at the next term or...divide a2 by a1 to find the common ratio, 2/3.

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Formulas used with geometric sequences and geometric series:

To find any term of a geometric sequence:

where a1 is the first term of the sequence,

r is the common ratio, n is the number of the term to find.

Note: a1 is often simply referred to as a.

To find the sum of a certain number of terms of a geometric sequence:

where Sn is the sum of n terms (nth partial sum),

a1 is the first term, r is the common ration.

Examples:

Question Answer 1. Find the common ratio for the sequence

1. The common ratio, r, can be found by dividing the second term by the first term, which in this problem yields -1/2. Checking shows that multiplying each entry by -1/2 yields the next entry.

2. Find the common ratio for the sequence given by the formula

2. The formula indicates that 3 is the common ratio by its position in the formula. A listing of the terms will also show what is happening in the sequence (start with n = 1). 5, 15, 45, 135, ... The list also shows the common ratio to be 3.

3. Find the 7th term of the sequence 2, 6, 18, 54, ...

3. n = 7; a1 = 2, r = 3

The seventh term is 1458.

4. Find the 11th term of the sequence

4. n = 11; a1 = 1, r = -1/2

5. Find a8 for the sequence 0.5, 3.5, 24.5, 171.5, ...

5. n = 8; a1 = 0.5, r = 7

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6. Evaluate using a formula:

6. Examine the summation

This is a geometric series with a common ratio of 3. n = 5; a1 = 3, r = 3

7. Find the sum of the first 8 terms of the sequence -5, 15, -45, 135, ...

7. The word "sum" indicates a need for the sum formula. n = 8; a1 = -5, r= -3

8. The third term of a geometric sequence is 3 and the sixth term is 1/9. Find the first term.

8. Think of the sequence as "starting with" 3, until you find the common ratio.

For this modified sequence: a1 = 3, a4 = 1/9, n = 4

Now, work backward multiplying by 3 (or dividing by 1/3) to find the actual first term. a1 = 27

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9. A ball is dropped from a height of 8 feet. The ball bounces to 80% of its previous height with each bounce. How high (to the nearest tenth of a foot) does the ball bounce on the fifth bounce?

9. Set up a model drawing for each "bounce". 6.4, 5.12, ___, ___, ___ The common ratio is 0.8.

Answer: 2.6 feet

Practice with Arithmetic and Geometric Sequences and Series

Solve the following problems dealing with Arithmetic and Geometric Sequences and Series.

Look carefully at each question to determine with "which" sequence you are working. You may wish to have your graphing calculator handy.

1.

Find the sum of the first 20 terms of the sequence 4, 6, 8, 10, ...

Answer

2.

Find the sum of the sequence -8, -5, -2, ..., 7 Answer

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3.

Find the 9th term of the sequence

Answer

4.

Evaluate this series using a formula:

Answer

5.

Insert three geometric means between 1 and 81.

Answer

6.

Find a6 for an arithmetic sequence where a1 = 3x+1 and d = 2x+6.

Answer

7.

Find t12 for a geometric sequence where t1 = 2 + 2i and r = 3.

Answer

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8.

A display of cans on a grocery shelf consists of 20 cans on the bottom, 18 cans in the next

row, and so on in an arithmetic sequence, until the top row has 4 cans. How many cans,

in total, are in the display?

Answer

9.

How many terms of the arithmetic sequence -3, 2, 7, ... must be added together for the sum of the

series to be 116? Answer

10.

Find the indicated sum

Answer

11.

Given the sequence -4, 0, 4, 8, 12, ..., Darius devises a formula for the sum of n terms of the

sequence. His formula is

Is this formula correct? Show work to support your answer.

Answer

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Practice with Arithmetic and Geometric Sequences Word Problems

Solve the following problems dealing with Arithmetic and Geometric Sequences and Series.

Look carefully at each question to determine with "which" sequence you are working. You may wish to have your graphing calculator handy.

1.

You visit the Grand Canyon and drop a penny off the edge of a cliff. The distance the penny will fall is 16 feet the first second, 48 feet the

next second, 80 feet the third second, and so on in an arithmetic sequence. What is the total

distance the object will fall in 6 seconds?

Answer

2. The sum of the interior angles of a triangle is 180º, of a quadrilateral is 360º and of a pentagon is 540º. Assuming this pattern continues, find the

sum of the interior angles of a dodecagon (12 sides).

Answer

3. After knee surgery, your trainer tells you to return to your jogging program slowly. He

suggests jogging for 12 minutes each day for the first week. Each week thereafter, he suggests that you increase that time by 6

minutes per day. How many weeks will it be before you are up to jogging 60 minutes per

day?

Answer

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4.

You complain that the hot tub in your hotel suite is not hot enough. The hotel

tells you that they will increase the temperature by 10% each hour. If the

current temperature of the hot tub is 75º F, what will be the temperature of the hot tub

after 3 hours, to the nearest tenth of a degree?

Answer

5.

A culture of bacteria doubles every 2 hours. If there are 500 bacteria at the beginning, how many bacteria will there be after 24 hours?

Answer

6. A mine worker discovers an ore sample containing 500 mg of radioactive material. It is discovered that the radioactive material has a half life of 1 day. Find the amount of

radioactive material in the sample at the beginning of the 7th day.

Answer

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How did we get these formulas?

Arithmetic Formulas

The origin of the formula to find a specific term of an arithmetic sequence where the common difference between terms is d can be seen by examining the sequence pattern.

Notice that the coefficient of d is one less than the location of the term.

Thus we have

If we carry this idea further, we can find the formula for partial sums of an arithmetic sequence.

First examine the terms, as we did before, starting with the first term.

Now, try the same approach starting with the last term.

Now, add these two equations and notice values that disappear. We will end up with:

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Geometric Formulas

The origin of the formula to find a specific term of an geometric sequence where the common ratio is r can also be seen by examining the sequence pattern.

Notice that the exponent of r is one less than the location of the term.

Thus we have

If we carry this idea further, we can find the formula for partial sums of an geometric sequence.

First express the series as we did above.

Now, multiply both sides by the common ratio, r.

Now, subtract these two equations and notice values that disappear. We will end up with:

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