Arithmetic and Geometric Sequences (11.2)

Common difference

Common ratio

A sequence

Give the next five terms of the sequence for

2, 7, 12, 17, …

What is the pattern for the terms?

A sequence

Give the next five terms of the sequence for

2, 7, 12, 17, 22, 27, 32, 37, 42

This is an example of a sequence– a string of numbers that follow some pattern.

What’s our pattern here?

A sequence

Give the next five terms of the sequence for

2, 7, 12, 17, 22, 27, 32, 37, 42

What’s our pattern here? We add five to a term to get the next term.

When we add or subtract to get from one term to the next, that’s an arithmetic sequence.

Another sequence

Find the next five terms in this sequence?

8, 4, 2, …

What’s our pattern this time?

Another sequence

Find the next five terms in this sequence?

8, 4, 2, 1, .5, .25, .125, .0625

What’s our pattern this time? We divide each term by 2 to get the next term. (This is also multiplying by ½.)

When we multiply or divide to get the next term, we have a geometric sequence.

Terminology

We label terms as tn, where n is the place the term has in the sequence.

The first term of a sequence is t1.

So in the arithmetic sequence, t1 = 2.

In the geometric sequence, t1 = 8.

Terminology

We label terms as tn, where n is the place the term has in the sequence.

The second term of a sequence is t2.

The third is t3. Get it?

If the current term is tn, then the next term is tn+1. The previous term is tn-1.

Terminology

We list sequences in the abstract as

t1, t2, t3, … tn.

This is true whether the sequence is arithmetic, geometric, or neither.

Arithmetic sequence formula

If the pattern between terms in a sequence is a common difference, the sequence is

arithmetic, and we call that difference d.

tn = t1 + (n-1) d

(In other words, find the nth term by adding (n-1) d’s to the first term.)

Test it with our first sequence.

Arithmetic sequence formula

If the pattern between terms in a sequence is a common difference, the sequence is

arithmetic, and we call that difference d.

tn = t1 + (n-1) d

We can use this to find the first term, nth term, the number of terms, and the difference.

Geometric sequence formula

If the pattern between terms in a sequence is a common ratio, then it is a geometric sequence and we call that ratio r.

tn = t1rn-1

(In other words, find the nth term by multiplying t1 by r and do that (n-1) times.)

Test it with our second sequence.

Geometric sequence formula

If the pattern between terms in a sequence is a common ratio, then it is a geometric sequence and we call that ratio r.

tn = t1rn-1

We can use this to find the first term, the nth term, the number of terms, and the common ratio.

Sequence #3

Give the first five terms of the sequence for

t1 = 7

tn+1 = tn – 3

What is the pattern for the terms? Is this arithmetic or geometric? What is the tenth term?

Sequence #3

Give the first five terms of the sequence for

7, 4, 1, -2, -5

What is the pattern for the terms? We subtract 3 from a term to get the next one.

It is an arithmetic sequence.

The tenth term is t10 = 7 + (10-1) (-3) = -20.

Sequence #4

Find which term 101 is in the arithmetic sequence with t1 = 5, and d = 3.

Sequence #4

Find which term 101 is in the arithmetic sequence with t1 = 5, and d = 3.

101 = 5 + (n – 1)3101 = 5 + 3n – 3101 = 2 + 3n99 = 3nn = 33

So, the 33rd term.

Sequence #5

Find the 9th term of the sequence 1, -2, 4, …

What type of sequence is this?

What formula do we use?

Sequence #5

Find the 9th term of the sequence 1, -2, 4, …

What type of sequence is this?

Geometric, with a common ratio of -2.

What formula do we use? tn = t1rn-1

So, t9 = 1(-2)9-1 = (-2)8 = 256.

Sequence #6

Find which term 1536 is in the geometric sequence with t1 = 3, and a common ratio of 2.

Sequence #6

Find which term 1536 is in the geometric sequence with t1 = 3, and a common ratio of 2.

1536 = 3(2)n-1

512 = (2)n-1

(Ooh, want an exponent, need to use logs.)

n -1 = log2512 = log 512/ log2 = 9

n = 10

Sequence #Last

Find the 9th term of the sequence

1, 1, 2, 3, 5, 8, …

What type of sequence is this?

What formula do we use?

How do we graph it?