•A geometric sequence is one in which the ratio of successive terms is constant. In other words, each number in the sequence is multiplied by a constant number to obtain the next number in the sequence:

–A basic example of a geometric sequence is 1, 2, 4, 8, 16, …

Consider a geometric sequence in which the general terms are a1, a2, a3, …

In order to determine the ratio r of successive terms, simply use the formula r = an+1/an.

In general, the formula for the nth term of a geometric sequence is an = a1 rn-1.

–Example: Consider the sequence: 1, 3, 9, 27, … Find the 6th term of the sequence:

Notice that in this geometric sequence the ratio r = 9/3 = 3.

Also note that a1 = 1 and n = 6.

Therefore, a6 = 1 (3)6-1 = 35 = 243.

Combinations are a probability concept that should be understood well before test day.

•A combination of a group of objects is a selection of objects in which the order is not relevant.

–Let n be the total number of objects and r the number of available slots.

Then the total number of combinations where a set of r elements is selected from among a set of n elements is denoted nCr.

nCr = n!/r!x(n-r)!

–Example: Find the number of committees of three people that can be formed from five people.

5C3 = 5!/3!x(5-3)! = 5x4x3x2x1/3x2x1x2x1 = 5×4/2×1 = 10

A basic concept of probability that appears frequently on the SAT Exam is that of the mean or average of a set of data.

•To determine the average, also known as an arithmetic mean, of a set of n numbers, simply add all the numbers and divide by n.

–Example: Find the mean of 34, 21, and 15.

Find the sum. 34 + 11 + 15 = 60

Divide by n = 3. 60/3 = 20.

•If the average of a set of numbers is known and all but one number is given, determine the missing number using the formula below.

–Sum of n numbers = Average x n

–Example: If the average of 3 numbers is 10 and two of the numbers is 10 and 5, find the third number.

Average x n = (10) (3) = 30

Sum of n numbers = 10 + 5 + x = 15 + x

Missing number: 30 = 15 + x  à x = 15

Understand the slope of a line and how it can be used to solve various problems on the SAT Exam is a crucial concept to master.

•The slope, or steepness, of a line is a concept that will appear often in SAT problems. To find the slope, pick any two points on the line and apply the Slope Formula.

–Slope = Vertical Change / Horizontal Change = y2-y1/x2-x1

–Example: Find the slope of the line that contains the points (3, 4) and (1, 2).

Slope = 4-2/3-1 = 2/2 = 1

•A line that rises from left to right has a positive slope, while a line that falls from left to right has a negative slope.

–A horizontal line has a slope of zero, while a vertical line has a slope that is not defined.

•Note: Any two points on a given line can be used to determine the slope of the overall line.

•Remember that parallel lines have equal slopes.

•Also remember that perpendicular lines have slopes that are negative reciprocals of one another.

–Example: Two lines whose slopes are 2 and –½ are perpendicular to each other since their slopes are the negative reciprocals of each other.

Understanding the mathematical concepts behind inscribed circles and polygons is a necessary concept to master before taking the exam.

•A polygon is said to be inscribed in a circle if all its vertices are points on the circle.

–Note: If the polygon inscribed in the circle is a rectangle, then the diagonals of the rectangle are diameters of the circle.

•A circle is said to be inscribed in a polygon if the circle touches each side of the polygon at exactly one point.

–Note: If the polygon, in which a circle is inscribed, is a square, then each side of the square has the same length as the diameter of the circle.

•A common problem type on SAT Exams is to find the area of shaded regions that involve inscribed circles and polygons. The strategy to solve these types of problems is to first find the area of the outer region, and then subtract the portion that is not shaded.

•Note: Any line tangent to a circle is perpendicular to a radius that is drawn to the point of contact.

The SAT Exam involves many questions that challenge the student with solving linear equations. The process is simple if you follow some simple rules.

•A linear equation is an equation in which the highest exponent of a variable is 1.

•To solve a linear equation, perform the same arithmetic operations to both sides of the equation until the variable is isolated.

–Example: Solve 4x + 3 = 10.

Subtract 3 from both sides of the equation.

4x + 3 – 3 = 10 – 3  à 4x = 7

Divide both sides of the equation by 4.

4x/4 = 7/4  à x = 7/4

•Note: If the equations contain parentheses, first use the distributive law to remove them.

•Note: If the equation contains fractions, first multiply both sides of the equation by the LCD of the fractions to remove them.

On the SAT Exam, many questions involve the comparison of two fractions and the determination of which fraction is larger.

•When comparing fractions, use the following rules:

–If two fractions have the same denominator, the fraction with the greater numerator is the larger one.

•Example: 6/13 > 1/13

–If two fractions have the same numerator, the fraction with the greater denominator is the smaller one.

•Example: 3/8 < 3/4

–If two fractions have different numerators and denominators, convert both to decimals and compare the decimals. To write a fraction as a decimal, use a calculator to divide the denominator of the fraction into the numerator.

•Example: 1/2 < 3/4 since 0.50 < 0.75

•Note: When multiplying a decimal by a power of 10, simply move the decimal point one place to the right for each zero in the power of 10.

•Note: When dividing a decimal by a power of 10, simply move the decimal point one place to the left for each zero in the power of 10.