SAT Arithmetic and Geometric Sequences: Recognizing Patterns and Finding Terms

An arithmetic sequence is a list of numbers with a constant difference between consecutive terms. The sequence 2, 5, 8, 11, ... has a common difference (d) of 3. The formula for the nth term of an arithmetic sequence is a_n=a_1+(n-1)d where a_1 is the first term and d is the common difference. For the sequence 2, 5, 8, 11, ..., a_10=2+(10-1)*3=2+27=29. The sum of an arithmetic series (sum of the first n terms) is S_n=n/2*(a_1+a_n) or S_n=n/2*(2*a_1+(n-1)d). The sum of the first 10 terms of the sequence above is S_10=10/2*(2+29)=5*31=155. On the SAT, identifying that a sequence is arithmetic (look for constant difference) immediately tells you which formula to use. Once you identify the common difference, finding any term or the sum becomes a matter of substituting into the formula.

Recognizing arithmetic sequences by observing differences is a key skill. Given 10, 7, 4, 1, -2, the differences are -3, -3, -3, so this is arithmetic with d=-3. These patterns appear in word problems about arranged objects, payments, or distances traveled, so learning to recognize them in diverse contexts helps you apply the formulas correctly.

A geometric sequence has a constant ratio (r) between consecutive terms. The sequence 2, 6, 18, 54, ... has a common ratio of 3. The formula for the nth term is a_n=a_1*r^(n-1). For the sequence above, a_10=2*3^(10-1)=2*3^9=2*19683=39366. The sum of the first n terms is S_n=a_1*(1-r^n)/(1-r) when r≠1. The sum of the first 10 terms is S_10=2*(1-3^10)/(1-3)=2*(1-59049)/(-2)=59049-1=59048. Identifying a geometric sequence by looking for a constant ratio (divide consecutive terms) is the first step to applying the formula. Geometric sequences grow or shrink exponentially, so they model situations like bacterial growth, radioactive decay, or investments with compound interest.

Special case: infinite geometric series. If |r|<1, the series converges and has a sum: S_infinity=a_1/(1-r). For instance, 1+0.5+0.25+0.125+... (with r=0.5) sums to 1/(1-0.5)=2. This concept appears rarely on the SAT but is worth knowing if you encounter a series question asking for the sum of infinitely many terms.

Some SAT sequences are neither arithmetic nor geometric but follow other patterns. A sequence might be defined recursively, where each term depends on the previous term(s) according to a rule. For instance, the Fibonacci sequence (1, 1, 2, 3, 5, 8, ...) is defined as a_n=a_(n-1)+a_(n-2), where each term is the sum of the two preceding terms. To find a specific term in a recursive sequence, you either continue applying the rule term by term or look for a pattern that allows you to skip ahead. When encountering a sequence on the SAT, your first step is to compute a few differences or ratios to determine if it is arithmetic or geometric. If neither, look for the pattern rule by examining how consecutive terms relate. Once you identify the pattern, you can find any term or sum.

Series questions might present a sequence and ask for the sum of the first n terms or the nth term itself. Identifying the sequence type determines which formula to use. Always verify your pattern assumption by checking that it holds for a few terms before applying the formula to find the answer.

A 1-week sequence drill plan builds fluency. Day 1: Given sequences, identify whether each is arithmetic, geometric, or neither. Day 2: Find a common difference or ratio for arithmetic and geometric sequences. Days 3-5: Calculate the nth term and the sum of the first n terms for various sequences. Day 6: Solve word problems describing sequences (arranged objects, travel distance, money growth). Day 7: Review any errors and identify which sequence types or formulas were problematic. Common mistake: confusing the formula for arithmetic sum (using average) with the formula for geometric sum (using ratio). Drilling both until they feel distinct prevents this error. Sequences appear regularly on the SAT, so building automation through practice pays off.

On test day, when you encounter a sequence question, identify the type, extract the relevant parameters (first term, common difference/ratio, number of terms), select the appropriate formula, and substitute. If the sequence does not fit arithmetic or geometric patterns, look for a recursive rule or pattern that applies. Always check your answer for reasonableness by verifying that the formula works for a few known terms in the sequence. This quick verification catches errors in identifying the sequence type or calculating parameters.