Quick Answer
A geometric sequence is a numerical list where each term is found by multiplying the previous term by a constant ratio. On the Digital SAT, these concepts typically appear in the Math section (Module 1 or 2) approximately once per test, usually within word problems involving exponential growth or decay.
A geometric sequence is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio (r). The n-th term is defined by the formula a_n = a_1 * r^(n-1).
Question: In a geometric sequence, the first term is 5 and the second term is 15. What is the 4th term? Solution: 1. Find the common ratio (r): r = 15 / 5 = 3. 2. Use the formula a_n = a_1 * r^(n-1). 3. a_4 = 5 * 3^(4-1) = 5 * 3^3. 4. a_4 = 5 * 27 = 135. The 4th term is 135.
Confusing it with arithmetic sequences: Students may add a constant value instead of multiplying by a common ratio to find the next term.
Miscalculating the exponent: Using 'n' instead of 'n-1' in the general formula, which leads to calculating the term one position too far in the sequence.
Incorrect ratio identification: Dividing the first term by the second term instead of the second by the first, resulting in the reciprocal of the actual ratio.
Students targeting 750+ should know that the Digital SAT often presents geometric sequences as discrete versions of exponential functions (y = ab^x). Recognizing that the common ratio 'r' is equivalent to the base 'b' in an exponential function allows you to use the Desmos graphing calculator more effectively to solve for unknown terms or growth rates by looking for the intersection of the function and the term index.
Arithmetic Sequence
An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. On the Digital SAT, this concept appears in the Math section, typically within Module 2 as a more advanced linear relationship question. It tests a student's ability to identify patterns and apply the nth term formula to find specific values.
Exponent
An exponent indicates the number of times a base is multiplied by itself. On the Digital SAT, exponent rules are a fundamental component of the Math section, appearing in approximately 10-15% of questions. Students typically encounter these within the Passport to Advanced Math and Heart of Algebra categories.
Exponential Function
An exponential function is a mathematical relationship where a constant ratio determines the change in the dependent variable. On the Digital SAT, these functions frequently appear in the Math section, specifically within the Advanced Math domain, often requiring students to interpret growth or decay constants in real-world modeling word problems.
Ratio
A ratio is a mathematical comparison of two quantities by division, often expressed as a:b. On the Digital SAT, ratios are frequently tested in the Math section under Problem Solving and Data Analysis. These questions typically require students to scale quantities or determine proportional relationships in word problems or data tables.
Sequence
A sequence is an ordered list of numbers following a specific rule. On the Digital SAT, sequences appear in the Math section, typically within Advanced Math. These questions occur approximately once or twice per test, requiring students to identify patterns or calculate specific terms using arithmetic or geometric formulas.
A geometric sequence on the Digital SAT is a set of numbers where each subsequent value is determined by multiplying the previous value by a fixed constant ratio. These sequences are foundational to understanding exponential growth and decay. In the Math section, you will typically see these as discrete models or word problems where you must identify the growth factor or calculate a specific future term.
To calculate terms in a geometric sequence, first identify the common ratio (r) by dividing any term by its preceding term. Once you have the first term (a1) and the ratio, use the general formula a_n = a_1 * r^(n-1) to find the n-th term. On the SAT, you can also use the Desmos calculator to iterate the multiplication if the term number is small.
The primary difference between a geometric sequence and an arithmetic sequence is the operation used to find the next term. In a geometric sequence, you multiply the current term by a constant ratio to get the next one, creating exponential change. In an arithmetic sequence, you add a constant difference to the current term, resulting in a linear progression with a constant rate of change.
Geometric sequence questions appear relatively sparingly on the Digital SAT, typically appearing approximately once or twice per full-length practice test or official exam. They are most commonly found in the Advanced Math category of the Math modules. While infrequent, mastering them is essential for high scorers because they overlap significantly with exponential function concepts, which appear much more frequently.
