Arithmetic vs. Geometric Sequences: Recognizing Patterns and Finding Terms on the SAT

An arithmetic sequence adds the same value each term. Example: 2, 5, 8, 11, ... has common difference d=3 (each term is 3 more than the previous). The general formula is a_n=a_1+(n-1)d, where a_1 is the first term, d is the common difference, and a_n is the nth term. To find the 10th term: a_10=2+(10-1)·3=2+27=29. The sum of an arithmetic series (adding all terms) is S_n=n(a_1+a_n)/2 or S_n=n·a_1+n(n-1)d/2. These formulas are essential; memorize them and practice using them until calculations become automatic.

Identify arithmetic sequences by checking if consecutive differences are equal. If the sequence has constant addition/subtraction, it is arithmetic. Build a three-step identification routine: (1) calculate the difference between consecutive terms, (2) check if all differences match, (3) identify it as arithmetic if they do.

A geometric sequence multiplies by the same value each term. Example: 2, 6, 18, 54, ... has common ratio r=3 (each term is 3 times the previous). The general formula is a_n=a_1·r^(n-1). To find the 5th term: a_5=2·3^(5-1)=2·81=162. The sum of a geometric series is S_n=a_1(1-r^n)/(1-r) if r≠1. These formulas differ from arithmetic, so keep them straight: arithmetic uses addition (d), geometric uses multiplication (r). Identifying which type allows you to use the correct formula immediately.

Identify geometric sequences by checking if consecutive ratios are equal. Calculate each term divided by the previous term; if all ratios match, it is geometric. Practice identification until you recognize patterns instantly.

Before solving, identify whether a sequence is arithmetic or geometric by finding the difference (arithmetic) or ratio (geometric) between consecutive terms. Once identified, apply the correct formula. Common errors: (1) using arithmetic formulas for geometric sequences or vice versa, (2) misidentifying the pattern, (3) making arithmetic mistakes in calculating common difference or ratio. Build a verification routine: identify the pattern, apply the correct formula, check your answer by calculating it manually for one or two terms, then solve the full problem.

Daily drill: solve five sequence problems daily, forcing yourself to identify the pattern first, then solve. Time yourself aiming for 1 minute per problem. This speed and accuracy prevent careless errors on test day.

Some geometric series (sequences with many terms) converge to a finite sum if |r|<1 (the common ratio is between -1 and 1). The infinite sum formula is S∞=a_1/(1-r) if |r|<1. For example, 1+1/2+1/4+1/8+... has a_1=1 and r=1/2, so S∞=1/(1-1/2)=2. This formula appears on some SAT problems; memorize it. If |r|≥1, the series diverges (does not converge to a finite sum). Recognize this and answer "diverges" or "undefined sum" for such series. Understanding convergence separates strong SAT math students from average ones.

Practice identifying whether an infinite geometric series converges by checking |r|. Calculate the infinite sum for five examples daily. Build fluency so convergence recognition becomes automatic.