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.
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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.