Arithmetic and Geometric sequences

Chapter 13 - Sequences and Series Section 13.1 Arithmetic and Geometric Sequences

Definitions: (yes, that's right, this is important, know these!)

A sequence is a set of numbers, called terms, arranged in some particular order.

An arithmetic sequence is a sequence with the difference between two consecutive terms constant. The difference is called the common difference.

A geometric sequence is a sequence with the ratio between two consecutive terms constant. This ratio is called the common ratio.

Let's play guess the sequence!: I give you a sequence and you guess the type. No fair peeking at the answers. They are somewhere else on the page. (well hidden of course!) 1) 3, 8, 13, 18, 23, . . . 2) 1, 2, 4, 8, 16, . . . 3) 24, 12, 6, 3, 3/2, 3/4, . . . 4) 55, 51, 47, 43, 39, 35, . . . 5) 2, 5, 10, 17, . . . 6) 1, 4, 9, 16, 25, 36, . . . Got them all figured out? You can check them at the bottom of the page. Go on if they are correct. Review if you goofed. (Be honest out there!)

This is important! (how can you guess?)

Arithmetic formula: t_n = t₁+ (n - 1)d

t_nis the nth term, t₁ is the first term, and d is the common difference.

Geometric formula: t_n = t₁ ^. r^{(n - 1)}

t_n is the nth term, t₁ is the first term, and r is the common ratio.

Here's the answers to the 6 questions up stairs.

1) Arithmetic, the common difference d = 5

2) Geometric, the common ratio r = 2

3) Geometric, r = 1/2

4) Arithmetic, d = -4

5) Neither, why? (How about no common difference or ratio!)

6) Neither again! (This looks familiar, could it be from geometry?)

Sample problems:

Find the first four terms and state whether the sequence is arithmetic, geometric, or neither.

1) t_n = 3n + 2 To find the first four terms, in a row, replace n with 1, then 2, then 3 and 4

Answer: 5, 8, 11, 14 The sequence is arithmetic! d = 3

2) t_n = n² + 1 To find the first four terms, do the same as above!

Answer: 2, 5, 10, 17 The sequence is neither. Why?

3) t_n = 3^.2ⁿ Ditto for this one ( got it by now?)

Answer: 6, 12, 24, 48 The sequence is geometric with r = 2

Find a formula for each sequence.

1) 2, 5, 8, 11, 14, . . .

Work: It is arithmetic! So use the arithmetic formula you learned above!

t₁ = 2, look at the first number in the sequence!

d = 3, look at the common difference!

Therefore, t_n = 2 + (n - 1)3 and simplifying yields : t_n = 3n -1 ( tada!)

Try putting in 1, then 2, then 3, etc. and you will get the sequence!

2) 4, 8, 16, 32, . . .

Work: It is geometric! So use the geometric formula you learned up yonder!

t₁ = 4, look at the first number in the sequence!

r = 2, look at the common ratio!

Therefore, t_n = 4 ^. 2^{(n - 1)} and simplifying gives us: t_n = 2^.2ⁿ (Yikes stripes! Where did this come from. rewrite 2^{(n - 1)} as 2ⁿ ^. 2^-1 and cancel with the four!)

Try putting in 1, 2, 3, etc and see if you get the sequence back!

3) 21, 201, 2001, 20001, . . .

Work: Bummer! It's not geometric or arithmetic. What do I do now? Mr. K didn't give me a formula! Don't panic! Use your head and think!

Think of the sequence as (20 +1), (200+1), (2000 + 1), (20000 + 1), . . .

Then as this sequence:[(2)(10) +1],[(2)(100) +1], [(2)(1000) +1], [(2)(10000) +1]

Wait! Hold on here! I see a pattern! Cool, without a formula! Powers of 10!

How does this grab ya! t_n = 2^.10ⁿ + 1 Does this work? Try it and see!

Find the indicated term of the sequence.

1) sequence is arithmetic with t₁ = 5 and t₇ = 29. Find t₅₃

Work: Use the formula! 29 = 5 + 6d Where oh where did I get that! Substitution!

24 = 6d means d = 4
t₅₃ = 5 + 52^.4 = 213

2) Find the number of multiples of 9 between 30 and 901.

Work: What's the first multiple of 9 in the range? How about 36.

What's the last multiple of 9 in the range? How about 900.

Use the formula: 900 = 36 + 9(n - 1) and solve for n!

864 = 9n - 9

873 = 9n

97 = n There are 97 multiples in the range!

Well, you have a pretty good background now in these, so let's head for section 13-2. Recursive Definitions.