seqences and exponential functions

Part 1: Sequences

Arithmetic Sequences

An arithmetic sequence adds or subtracts the same number each time.
Common Difference (d): the amount added or subtracted.
Example: 4, 7, 10, 13
Common difference = +3

Geometric Sequences

A geometric sequence multiplies or divides by the same number each time.
Common Ratio (r): the number multiplied each time.
Example: 2, 6, 18, 54
Common ratio = ×3

Explicit Formula for Arithmetic Sequences

Formula: aₙ = a₁ + d(n − 1)
a₁ = first term
d = common difference
n = term number
Example: 5, 9, 13, 17
a₁ = 5, d = 4

Recursive Formula

A recursive formula tells the first term and how to find the next term.
Example:
a₁ = 3
aₙ = 3aₙ₋₁

Finding Terms in a Sequence

1. Identify the pattern.
2. Decide if it is arithmetic or geometric.
3. Use the formula or continue the pattern.

Part 2: Exponential Functions

Exponential Growth

Growth happens when the multiplier is greater than 1.
Formula: y = a(b)^x
a = starting amount
b = growth factor
Example: y = 150(2)^x

Exponential Decay

Decay happens when the multiplier is between 0 and 1.
Example: y = 20000(0.88)^x

Percent Growth and Decay

Growth: add the percent to 1.
Example: 25% growth → 1.25

Decay: subtract the percent from 1.
Example: 12% decay → 0.88

Solving Exponential Problems

1. Identify the starting amount.
2. Convert the percent to a decimal.
3. Decide if it is growth or decay.
4. Substitute into y = a(b)^x

Graphing Exponential Functions

Make a table of x-values and substitute them into the equation.
Example for y = 2^x:
x = -2, -1, 0, 1, 2

Quick Tips

• Arithmetic = add/subtract

• Geometric = multiply/divide

• Growth factor > 1

• Decay factor is between 0 and 1

• Always check the pattern before choosing a formula

• Negative exponents make fractions

Common Mistakes to Avoid

• Mixing up arithmetic and geometric sequences

• Forgetting to subtract 1 in arithmetic formulas

• Using the percent instead of the decimal multiplier

• Confusing growth and decay