A sequence is an ordered list of numbers

following a specific pattern or rule

Example: 2, 4, 6, 8, 10, ...

Term: Each number in a sequence

Position (n): The place of a term in the sequence

First term (a₁): The starting number

nth term (aₙ): The value of any term at position n

Arithmetic Sequence: Add or subtract the same number

Geometric Sequence: Multiply or divide by the same number

Neither: No consistent pattern

An arithmetic sequence is a sequence where

each term is found by ADDING (or subtracting)

the SAME number to the previous term

Examples: 3, 7, 11, 15, 19, ... (add 4)

Examples: 20, 17, 14, 11, 8, ... (subtract 3)

The common difference is the number

added to each term to get the next term

d = a₂ − a₁

or

d = aₙ − aₙ₋₁

Subtract any term from the next term

aₙ = a₁ + (n − 1)d

Where:

aₙ = nth term (the term you want to find)

a₁ = first term

n = position of the term

d = common difference

Sequence: 5, 9, 13, 17, 21, ...

Find the 10th term of: 3, 7, 11, 15, ...

Write an expression for: 2, 5, 8, 11, 14, ...

A geometric sequence is a sequence where

each term is found by MULTIPLYING (or dividing)

the previous term by the SAME number

Examples: 2, 6, 18, 54, 162, ... (multiply by 3)

Examples: 80, 40, 20, 10, 5, ... (divide by 2)

The common ratio is the number

multiplied to each term to get the next term

r = a₂ ÷ a₁

or

r = aₙ ÷ aₙ₋₁

Divide any term by the previous term

aₙ = a₁ × rⁿ⁻¹

Where:

aₙ = nth term

a₁ = first term

n = position of the term

r = common ratio

Sequence: 3, 12, 48, 192, ...

Find the 6th term of: 2, 6, 18, 54, ...

Sequence: 80, 40, 20, 10, 5, ...

Step 1: Test for arithmetic (subtract consecutive terms)

If differences are the same → ARITHMETIC

Step 2: Test for geometric (divide consecutive terms)

If ratios are the same → GEOMETRIC

Step 3: If neither pattern works → NEITHER

Sequence: 5, 10, 15, 20, 25, ...

Sequence: 4, 12, 36, 108, ...

Sequence: 1, 1, 2, 3, 5, 8, ... (Fibonacci)

Given a formula for a sequence,

substitute the term position (n) to find the value

Given aₙ = 4n + 1, find a₅

Given aₙ = 3 × 2ⁿ⁻¹, find a₄

Problem: Sarah saves $10 in week 1, $15 in week 2, $20 in week 3. If this pattern continues, how much will she save in week 8?

Problem: A bacteria culture doubles every hour. It starts with 5 bacteria. How many bacteria will there be after 5 hours?

Finding d or r

✓ Arithmetic: Add/subtract → Use d (common difference)

✓ Geometric: Multiply/divide → Use r (common ratio)

✓ Test arithmetic first: Check if differences are constant

✓ Then test geometric: Check if ratios are constant

✓ Formula variables: a₁ = first term, n = position, aₙ = nth term

✓ For arithmetic: aₙ = a₁ + (n − 1)d

✓ For geometric: aₙ = a₁ × rⁿ⁻¹

✓ Variable expressions: Simplify to get cleanest form

✓ Word problems: Identify if pattern is arithmetic or geometric first

✓ Always check: Does your answer make sense in the sequence?

Arithmetic vs Geometric:

"Arithmetic adds with care, Geometric multiplies everywhere!"

Common Difference:

"D is for Difference - subtract to find, arithmetic patterns are one of a kind!"

Common Ratio:

"R is for Ratio - divide to see, geometric growth or decay it will be!"

Arithmetic Formula:

"Start with first, then (n minus 1) times d - that's arithmetic, guaranteed!"

Geometric Formula:

"First term times r to the power - geometric sequences gain or lose power every hour!"

Identifying Sequences:

"Subtract for arithmetic, divide for geometric too - if neither works, it's something new!"