seqences and exponential functions

Arithmetic Sequence

A sequence that adds or subtracts the same number each time.

Common Difference (d)

The amount added or subtracted in an arithmetic sequence.

Geometric Sequence

A sequence that multiplies or divides by the same number each time.

Common Ratio (r)

The number multiplied each time in a geometric sequence.

Explicit Formula for Arithmetic Sequences

aₙ = a₁ + d(n − 1), where a₁ is the first term, d is the common difference, and n is the term number.

Recursive Formula

A formula that provides the first term and method to find the next term.

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.

Exponential Growth

Growth occurs when the multiplier is greater than 1, represented by the formula y = a(b)^x.

Exponential Decay

Decay occurs when the multiplier is between 0 and 1, e.g., y = 20000(0.88)^x.

Percent Growth

To calculate growth, add the percent to 1; for example, 25% growth becomes 1.25.

Percent Decay

To calculate decay, subtract the percent from 1; for example, 12% decay becomes 0.88.

Steps for Solving Exponential Problems

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

Quick Tips for Sequences and Functions

Arithmetic means add/subtract; geometric means multiply/divide; growth factor is > 1; decay factor is between 0 and 1.

Common Mistakes to Avoid

Mixing arithmetic and geometric sequences; forgetting to subtract 1 in arithmetic formulas; using percent instead of decimal; confusing growth and decay.