Sequences – Definition of Sequence

• Sequence is a list of number of objects in a special order.

•Example:

•3, 5, 7, 9, …… is a sequence starting at 3 and increasing by 2 each time.

• A sequence is a (finite or infinite) set of numbers.

Example problem involving Sequence

Problem Number One:

If the first three terms of an arithmetic sequence are 2, 6 and 10, find the 40th term.

To solve the problem we use this formula for finding the nth term of an arithmetic sequence.

An = A + (n - 1) d

Where, An = is the nth term, in the case of our problem it is the 40th term

A = the first term of the sequence, in our problem it is 2.

n = number of terms, in our problem it is 40.

d = the interval of the terms, or the difference of the next term from the previous term,

to get d; d = 6 - 2 = 4.

Now, it is time to substitute the values to the formula for solving nth term where the 40th term is to be solved.

An = 2 + (40 - 1) 4

An = 2 + (39) 4

An = 2 + 156

An = 158.

The 40th term of the arithmetic sequence is 158.

Problem Number Two:

If the first term of an arithmetic sequence is -3 and the eighth term is 11, find d and write the first 10 terms of the sequence.

In this problem,

A = -3 n = 8 A8 = 11

If these values are substituted in the formula for An, we have

11 = -3 + (8 - 1) d

11 = -3 + 7d

14 = 7d

d = 2

The first ten terms are -3, -1, 1, 3, 5, 7, 9, 11, 13, 15

Arithmetic Progression

• A sequence in which the difference between any two successive terms is constant is called an arithmetic sequence @ arithmetic progression.
• The constant difference is called the common difference.
• Denoted by d.

Geometric Progression

• A sequence in which the ratio of every pair of successive terms is constant is called a geomatric sequence @ geometric progression
• The constant ratio is called the common ratio.
• Denoted by r.