
	Sequences and Series
	Rational Expressions
	Financial Mathematics
	Quadratic Functions and Complex Numbers
	Trigonometric Functions
	Graphing Trigonometric Functions

Arithmetic Sequences and Series

{ common difference, general term of sequence, finding sum }

Tharanky Balachandran

Overview

In an arithmetic sequence, you always add the same number to a term to get the next term. The number obtained by subtracting any term from the next term is a constant. This constant is called the common difference. The major concepts are the relationship and pattern between sequences and series. How do sequences and series function. As a standard question you are expected to know:

Ø How to find the common difference

Ø How to find the general term

Ø How to find values using the general term

Ø Determining the general term and other values when only given two values of two terms

Ø Finding the sum

Ø Determining sum values when only given two sums at different position

Study Tips, Methods and or Advice

General term for arithmetic sequences: tn = a + ( n � 1 ) d

Where tn = general term � the value of a number at a certain position in an arithmetic sequence

a = initial value

n = position of number in sequence

d = common difference

Equation for common difference: d = tn �tn-1

Example: finding general term for arithmetic sequence: 5, 10, 15, 20, 25

Where a = first term: 5

d = tn � tn-1

= t2 � t2-1

= t2 � t1

= 10 � 5

= 5

tn = a + ( n � 1 ) d

= 5 + ( n � 1) 5

= 5 + 5n � 5

= 5n

General Term: tn = 5n

Finding tn using general term:

Substitute the position of the number in that sequence to find the value of that number

So for the 10^th term of the sequence given the general term: tn = 5n

tn = 5n

t10 = 5(10)

t10 = 50

Therefore for the 10^th term of this sequence the value is 50.

Finding d when you are only given two numbers at different positions.

Example: t2 = 14

t5 = 23

1) Make chart and put in values:

n	1	2	3	4	5
tn		14			23

2) First given number + number of terms in between � d = second given number

14 + 3d = 23

3d = 23 �14

3d = 9

3d/3 = 9/3

d = 3

Therefore common difference is 3

Now you can use the common difference to figure out the missing value.

Finding the 1^st term :

d = tn � tn-1

3 = t2 � t2-1

3 = t2 � t1

3 = 14 � t1

11 = t1

1^st term is 11.

Finding sum of arithmetic sequences:

Two formulas:

1) Sn = ((a + tn) /2) n

2) Sn = n/2 (2a + (n-1) d)

Where:

Sn = total sum

n = position of number in sequence

a = initial value

d = common difference

Sample Questions and complete solutions

Questions

Complete Solutions

1. Write the general term for the following sequence:

4, 9, 14, 19, 24, �.

a = 4 tn = a + (n-1) d

d = 5 = 4 + (n-1) 5

= 4 + 5n � 5

= 5n � 1

2. Write the general term for the following sequence:

1/1, 4/3, 8/6, 12/9, 16/12, �

You do two general term formula:

Top : a = 1 tn = a + (n-1)

d = 3 = 1 + (n-1) 3

= 1 + 3n �3

= 3n � 2

Therefore tn = 3n � 2/ 2n - 1

Bottom: a =1 tn = a + (n-1) d

d = 2 = 1 + (n-1) 2

= 1 + 2n �2

= 2n � 1

3.Determine the first four terms of the sequence given the general term:

tn = 3n + 2

tn = 3n + 2 tn = 3n + 2 tn = 3n + 2 tn = 3n + 2

t1= 3(1) + 2 t2 = 3(2) + 2 t3 = 3(3) + 2 t4 = 3(4) + 2

= 3 + 2 = 6 + 2 = 9 + 2 = 12 + 2

= 5 = 8 = 11 = 14

First four terms are: 5, 8, 11, 14, ..

4.Determine the third term given that the second term is 5 and the fifth term is 11.

Fifth term � second term = 3 d

First given number + number of terms in between �d = second given number

5 + 3d = 11 d = tn � tn-1

5 + 3d = 11 �5 2 = t3 � t3-1

3d = 6 2 = t3 � t2

3d/3 = 6/3 2 = t3 - 5

d = 2 7 = t3

The third term is 7

5. Determine the sum of the series:

3 + 7 + 11 + 15 + �.. + 27

a = 3 tn = 4n � 1 Sn = n/2 (2a + (n-1) d)

d = 4 27 = 4n �1 = 7/2 (2(3) + (7-1) 4)

28 = 4n = 3.5 (6 + (6) 4)

tn = a + (n-1) d 28/4 = 4n/4 = 3.5 (6 + 24)

= 3 + (n �1) 4 7 = n = 3.5 (30)

= 3 + 4n � 4 = 105

= 4n �1 The sum of the series is 105.

6.Given that the 4^th term is 24 and the 8^thterm is 40.

a) Determine the value of d (common difference)

b) Determine the value of a (value of first term)

c) Determine the value of the 10^th term.

a) Find two equations by substituting values into general term:

tn = a + (n-1) d

24 = a + (4�1) d

24 = a + 3d

tn = a + (n-1) d

40 = a + (8-1) d

40 = a + 7d

Subtract the bigger value equation by the smaller value equation:

40 = a + 7d

- 24 = a + 3d

16 = 4d

16/4 = 4d/4

4 = d

The value of d is 4

b) Substitute the value of d in one of the new equations:

24 = a + 3d

24 = a + 3(4)

24 = a + 12

24 �12 = a

12 = a

The value of a is 12

c) Write the general term:

tn = a + (n-1) d

tn = 12 + (n-1) 4

tn = 12 + 4n � 4

tn = 4n + 8

t10 = 4(10) + 8

= 40 + 8

= 48

The value of the 10^th term is 48.

7. Given that the sum of the 4^th term is 20 and the 6^thterm is 32.

a) Determine the value of a (value of first term)

b) Determine the value of d (common difference)

c) Determine the sum of the 20^th term.

a) Find two equations by substituting values into the sum equation:

Sn = n/2 (2a + (n-1) d )

20 = 4/2 (2a + (4-1) d)

20 = 2 (2a + 3d)

20 = 4a + 6d

Sn = n/2 (2a + (n-1) d)

32 = 6/2 (2a + (6-1) d)

32 = 3 (2a + 5d)

32 = 6a + 15d

Take one of the new equations and solve for d (common difference):

20 = 4a + 6d

20 � 4a = 6d

(20 �4a)/6 = d

Substitute the value of d in one of the equations:

32 = 6a + 15d

32 = 6a + 15((20 � 4a)/6)

32 = 6a/6 +(300- 60a)/6)

32 = a + 50 � 10a

32 = 50 � 9a

32 �50 = -9a

-18 = -9a

-18/-9 = -9a/-9

2 = a

The value of a is 2.

b) Substitute the value found for a in part a) into one of the equations:

20 = 4a + 6d

20 = 4(2) + 6d

20 = 8 + 6d

20 � 8 = 6d

12 = 6d

12/6 = 6d/6

2 = d

The value of d is 2.

c) Write the sum equation and substitute values:

Sn = n/2 (2a + (n-1) d )

S20=20/2(2(2)+(20-1)2)

S20 = 10 ( 4 + (19) 2 )

S20 = 10 (4 +38)

S20 = 10 (42)

S20 = 420

The sum of the 20^th term is 420.

Extra Practice Questions and Answers

Extra Practice Questions

Answers

1.Write the general term for the series: 30, 26, 22, 18, 14, �

The general term is:

tn = 4n �34

2. Determine the first five terms of the following sequence: tn = 2n �23

The first five terms are:

-21, -18, -15, -12, -9

3. Determine the forth term given that the first term is �12 and the eighth term is 16.

The forth term is 0

4.Determine the sum of the series: 4 + 11 + 18 + ��.. +102

The sum of the series is 795

5.Determine the sum of the first 20 terms given:

2.5 + 4.5 + 6.5 + 8.5�.t20

The sum of the first 20 terms is 40

6. The 3^rd term of a series is �1 and the 7^th term is 2. What is the 10^th term?

The 10^th term is 4.25

Self Reflection

Personal Reflection - Tharanky

Arithmetic sequences and series are very simple to understand, but you have to remember the formulas and how to operate them. Sometimes you can solve these questions in other methods such as entering all these numbers in your calculator. Well that�s ok for just couple terms, but what happens when you have 50 terms. Then you�re in trouble! But I learned that in using these equations we could determine the values much quicker. There are some questions where you have to apply almost all the formulas, which is frustrating. But on the positive side it doesn�t take as much time as using a calculator. So therefore I think these formulas are very helpful. I enjoyed working on this assignment because in helping people to understand how to solve arithmetic series and/or sequence, I also helped myself understand the subject better as well. Thus it was a positive experience. I think this project is a fantastic idea because the material that these web pages contain is a excellent way for students to study for the exam by not only learning the concepts but they are also shown how to apply them and they are given sample questions to practice. Overall I think this assignment was not only enjoyable but also beneficial.