A geometric sequence is a list of numbers where you multiply by the same number each time to get the next term. That number is the common ratio, r. For example, 2, 6, 18, 54, … multiplies by 3. The formula aₙ = a₁ · rⁿ⁻¹ finds any term.
Compound interest is interest earned on both the money you put in and the interest it has already earned. The formula is A = P(1 + r/n)ⁿᵗ, where P is the starting amount, r is the yearly rate as a decimal, n is how many times a year interest is added, and t is the number of years.
Exponential growth and decay describe amounts that change by the same percent each time period. For growth, use y = a(1 + r)ᵗ. For decay, use y = a(1 − r)ᵗ. Here a is the starting amount, r is the rate as a decimal, and t is the number of time periods.
An exponential function has the variable in the exponent. Its form is f(x) = a · bˣ, where a is the starting value and b is the number you multiply by each time x goes up by 1. For example, f(x) = 3 · 2ˣ doubles at every step.
Zero and negative exponents follow two rules. Any nonzero number to the zero power is 1, so 7⁰ = 1. A negative exponent means “one over” the positive power, so x⁻ⁿ = 1/xⁿ. For example, 2⁻³ = 1/8. A negative exponent never makes the number negative.
Exponent rules are shortcuts for simplifying powers. When the bases match, multiply by adding the exponents: x^a times x^b = x^(a+b). Divide by subtracting them. For a power of a power, multiply the exponents. For a power of a product or quotient, give the exponent to each factor.
A system of inequalities is two or more inequalities with the same variables. Its solutions are all the points that make every inequality true. To solve it, graph each inequality on the same grid. The solution is the region where the shaded areas overlap.
Systems of equations word problems describe two unknown amounts with two facts. Name a variable for each unknown and write one equation for each fact. Then solve the system by substitution, elimination or graphing. Last, check that the answer makes sense in the story.
An arithmetic sequence is a list of numbers where you add the same amount each time to get the next number. That amount is the common difference, d. For example, 3, 7, 11, 15, … adds 4 each time. The formula a(n) = a(1) + (n – 1)d finds any term.
The average rate of change of a function tells how much the output changes, on average, for each 1 unit of input over an interval. To find it, divide the change in y by the change in x. It is the slope of the line through the two end points of the interval.