DrDelMath

Chapter 8:   Sequences, Induction, and Probability

Section 8.1: Sequences and Summation Notation (page 668)

Section 8.2: Arithmetic Sequences (page 679)

Definition: An arithmetic sequence is a sequence whose consecutive terms have a common difference.

Definition: The difference between consecutive terms of an arithmetic sequence is called the common difference of the sequence.

Equivalent Definition: An arithmetic sequence f is a function whose rule may be expressed as a linear equation of the form f(n) = dn + b where d is the common difference and b is the difference f(1) – d.

Fact: The n^th term of an arithmetic sequence named a is given by the rule a_n = dn + c where d is the common difference and c is a constant.
Observe that if we used normal functional notation the rule would be written as
a(n) = dn + c
which looks like the familiar linear function whose rule is usually presented as
f(x) = mx + b.

Comments:Compare the equivalent definition of an arithmetic sequence with the definition of a linear function to conclude the following.

The domain of a linear function is R
The domain of an arithmetic sequence is N

The rule for a linear function is f(x) = mx + b
The rule for an arithmetic sequence is a(x) = dx + b

The number b in the rule for an arithmetic sequence is the range value associated with 0 (if there were such a range element) so it corresponds exactly to the y-intercept of the linear function.

The common difference d is nothing more than the slope as you move from one range element to the next.

The slope of the line joining two terms (x, f(x)) and (x + 1, f(x + 1)) of the sequence f is given by.

A casual approach is to view an arithmetic sequence as a linear function with domain N.

Comment: Given any two pieces of information about an arithmetic sequence it is possible to determine its rule.  The next three problem types illustrate the point.

Problem Type 1: If you are given the common difference and the first term of the arithmetic sequence, then it is possible to write the rule for the function. This is comparable to the slope-intercept situation/problem when working with linear functions.

Example: Suppose an arithmetic sequence named h has a common difference 8 and the first term is -5. Find the rule for the function h.
Solution: Since the function is an arithmetic sequence its rule is of the form
h(n) = dn + b. In our case the common difference d is 8, so the rule for h has the form
h(n) =8n + b. Because the first term is -5, b = -5 – 8 = -13 and the rule for the desired arithmetic sequence is given by h(n) = 8n – 13.

Problem Type 2: If you are given the common difference d and one term of an arithmetic sequence, then it is possible to write the rule for the function. This is comparable to the point-slope situation/problem when working with linear functions.

Example: Suppose an arithmetic sequence named h has a common difference 3 and the fifth term is 12. Find the rule for the function h.
Solution: Since the function is an arithmetic sequence its rule is of the form
h(n) = dn + b. In our case the common difference d is 3, so the rule for h has the form
h(n) =3n + b. Because the fifth term is 12, h(5) = 12, but according to the partially determined rule h(5) = 3(5) + b = 15 + b. These two representations for h(5) yield the equation 12 = 15 + b. Clearly then b = -3 and the rule for the desired arithmetic sequence is given by
h(n) = 3n – 3.

Problem Type 3: If you are given two terms of an arithmetic sequence, then it is possible to write the rule for the function. This is comparable to the two point situation/problem when working with linear functions.

Example: Suppose the fourth term an arithmetic sequence named k is 10 and the seventh term is 28. Find the rule for the function h.

Solution: Since the function is an arithmetic sequence its rule is of the form
h(n) = dn + b. The difference between the seventh and fourth terms is 3d and is also equal to 28 – 10 = 18. That means 3d = 18 and so the common difference d is 6.

The rule for h has the form h(n) =6n + b. Because the fourth term is 10, h(4) = 10, but according to the partially determined rule h(4) = 6(4) + b = 24 + b. These two representations for h(4) yield the equation 10 = 24 + b. Clearly then b = -14 and the rule for the desired arithmetic sequence is given by h(n) = 6n – 14.

Comment: The formula for the nth partial sum of an arithmetic sequence named a is:

Section 8.3: Geometric Sequences (page 688)

Definition:A sequence a is a geometric sequence if the ratio of consecutive terms of a is the same for all terms of the sequence.

Definition: The ratio between consecutive terms of a geometric sequence is called the common ratio of the sequence.

Fact: The n^th term of a geometric sequence named a is given by the rule

a_n = a₁r^{n - 1}

where r is the common ratio.

Observe that if we used normal functional notation the rule would be written as

a(n) = a₁r^{n - 1}

which looks like a familiar exponential function whose rule is normally presented as
f(x) = ar^x.

Section 8.4: Mathematical Induction

Section 8.5: The Binomial Theorem

Good Explanation

Section 8.6: Counting Principles, Permutations, and Combinations

Section 8.7: Probability