Part 8: Sequences and Functions

# 8.1: Sequences

Notes

A sequence is an ordered list of entries, usually numbers, based on a repeating pattern.

##### Arithmetic sequences

The next entry in the sequence is the previous entry plus a fixed number.

**Example 1:**2, 6, 10, 14, …**Pattern:**Start at 2 and increase by steps of 4.- What is the next (5th) entry in the sequence? 14 + 4 = 18.
- What would be the “zero-th” (0th) entry (the one before the first entry)? 2 – 4 = –2.
- What is a
**formula**for finding the*n*th entry?- step x n + 0th entry
- i.e.,
*n*th entry = 4*n*– 2.

- Check: 1st entry = 4(1) – 2 = 2.
- Check: 5th entry = 4(5) – 2 = 18.
- What is the 100th entry? 4(100) – 2 = 398.

**Example 2:**2, –3, –8, –13, …**Pattern:**Start at 2 and increase by steps of –5.- What is the next (5th) entry in the sequence? –13 – 5 = –18.
- What would be the “zero-th” (0th) entry? 2 – (–5) = 7.
- What is a
**formula**for finding the*n*th entry?- step x n + 0th entry
- i.e.,
*n*th entry = –5*n*+ 7.

- Check: 1st entry = –5(0) + 7 = 7.
- Check: 5th entry = –5(5) + 7 = 18.
- What is the 100th entry? –5(100) + 7 = –493.

##### Geometric sequences

The next entry in the sequence is the previous entry multiplied by a fixed number.

**Example 1:**3, 3.6, 4.32, 5.184, …**Pattern:**Start at 3 and increase by repeatedly multiplying by 1.2.- What is the next (5th) entry in the sequence? 5.184 x 1.2 = 6.2208.
- What would be the “zero-th” (0th) entry (the one before the first entry)? 3 ÷ 1.2 = 2.5.
- What is a
**formula**for finding the*n*th entry?- multiplier
^{n}x 0th entry - i.e.,
*n*th entry = 1.2^{n}x 2.5.

- multiplier
- Check: 1st entry = 1.2
^{1}x 2.5 = 3. - Check: 5th entry = 1.2
^{5}x 2.5 = 6.2208. - What is the 100th entry? 1.2
^{100}x 2.5 = 207,044,936.

**Example 2:**57, 54.15, 51.4425, 48.870375, …**Pattern:**Start at 57 and decrease by repeatedly multiplying by 0.95.- What is the next (5th) entry in the sequence? 48.870375 x 0.95 = 46.42685625.
- What would be the “zero-th” (0th) entry (the one before the first entry)? 57 ÷ 0.95 = 60.
- What is a
**formula**for finding the*n*th entry?- multiplier
^{n}x 0th entry - i.e.,
*n*th entry = 0.95^{n}x 60.

- multiplier
- Check: 1st entry = 0.95
^{1}x 60 = 57. - Check: 5th entry = 0.95
^{5}x 60 = 46.42685625. - What is the 100th entry? 0.95
^{100}x 60 = 0.35523175.

The video below works through some examples of sequences.

Video Tips

Practice Exercises

Do the following exercises to practice working with sequences.