GMAT Sequences: Patterns and Formulas

A sequence is an ordered list of numbers, and GMAT Quantitative Reasoning turns a handful of these lists into short algebra problems. Each question gives you a starting term and a rule, then asks for a term far down the list or for the total of several terms. Two rules cover almost every version you meet on the exam: arithmetic sequences add a fixed difference between terms, and geometric sequences multiply by a fixed ratio between terms.

GMAT Quantitative Reasoning holds 21 Problem Solving questions in 45 minutes, and GMAC lists arithmetic and elementary algebra as the tested content, so sequence questions draw on the same algebra material. The GMAT Focus Edition removed geometry from Quantitative Reasoning entirely. No calculator, on-screen or personal, is allowed during this section, so every formula below needs to work by hand.

Where Sequences Sit in the GMAT

Quantitative Reasoning is one of three sections on the GMAT exam structure, which runs 64 questions across 2 hours 15 minutes, with one optional 10-minute break. Each section carries its own question count and its own 45-minute clock. No calculator is available in Quantitative Reasoning, so every method below is built for pen and paper. The table below breaks down each section.

Arithmetic Sequences

An arithmetic sequence adds the same number, the common difference, to each term to get the next term. The sequence 4, 9, 14, 19 adds 5 each time, so the common difference is 5. A negative common difference makes the sequence decrease instead of increase.

The nth-Term Formula

The nth term of an arithmetic sequence equals the first term plus (n minus 1) times the common difference: a_n = a_1 + (n - 1)d. The (n minus 1) counts steps rather than terms, so the first term takes zero steps of size d and the fifth term takes four. Count steps rather than terms and the off-by-one error disappears. The table below defines each symbol in the formula.

Common Difference and Term Position

Find the common difference by subtracting one term from the term after it. In the sequence 12, 7, 2, -3, each term drops by 5, so d = -5. When the two known terms are not adjacent, work backward to d and a_1. Follow the steps below in order.

1. Write the formula a_n = a_1 + (n - 1)d for each known term, using its own position number.
2. Subtract one equation from the other so a_1 cancels out, leaving an equation in d alone.
3. Solve for d, then substitute it back into either original equation to find a_1.
4. Use a_1 and d in the nth-term formula to compute the term the question asks for.

Geometric Sequences

A geometric sequence multiplies each term by the same number, the common ratio, to get the next term. The sequence 3, 6, 12, 24 multiplies by 2 each time, so the common ratio is 2. A ratio between 0 and 1 shrinks the sequence toward zero, and a negative ratio flips the sign of every other term.

The nth-Term Formula

The nth term of a geometric sequence equals the first term times the common ratio raised to the power of (n minus 1): a_n = a_1 x r^(n-1). The exponent counts steps the same way the arithmetic formula does, so the first term carries r^0, which equals 1. Multiplication replaces addition here, and the exponent on r replaces the multiplier on d. The table below defines each symbol.

Common Ratio and Term Position

Find the common ratio by dividing one term by the term before it. In the sequence 81, 27, 9, 3, each term divides by 3, so r = 1/3. The sequence 2, -6, 18, -54 uses r = -3, and the sign alternates with every step. When a question gives two non-adjacent terms of a geometric sequence, divide the later term by the earlier term to isolate a power of r, then take the matching root to solve for r itself.

Recursively Defined Sequences

A recursive sequence defines each term using the term or terms before it instead of a direct formula tied to position n. The definition names a starting value and a rule connecting each new term to the one (or two) before it. Without the starting value the rule produces nothing, so read the starting value first. Recursive rules build arithmetic and geometric sequences as well as patterns neither formula covers.

The rule a_1 = 3, a_n = a_(n-1) + 4 for n greater than 1 builds the sequence 3, 7, 11, 15, which matches an arithmetic sequence with d = 4. The rule a_1 = 2, a_n = 3 x a_(n-1) builds 2, 6, 18, 54, which matches a geometric sequence with r = 3. Some recursive rules pull from two previous terms at once. The rule a_1 = 1, a_2 = 1, a_n = a_(n-1) + a_(n-2) for n greater than 2 builds 1, 1, 2, 3, 5, 8, where every term from the third one onward equals the sum of the two terms before it.

When a recursive rule asks for a term a few steps ahead, list every term in order until you reach it. Write each step on paper instead of tracking it in your head, since one wrong term corrupts every term after it. A closed-form shortcut is not part of the definition, so the listing is the method rather than a fallback. Stop as soon as you hit the term the question names.

Sum-of-Sequence Shortcuts

Adding every term one by one wastes time on a timed section. These two formulas turn a long addition problem into a single calculation. Both need the number of terms, n, so count the terms before you apply either one. For an arithmetic sequence the term count equals (a_n - a_1) / d + 1.

Sum of an Arithmetic Sequence

The sum of the first n terms of an arithmetic sequence equals n times the average of the first and last term: S_n = (n/2)(a_1 + a_n). Pair the first term with the last term, the second term with the second-to-last term, and so on. Every pair adds to the same total, which is why the shortcut works. For the sequence 2, 5, 8, up to 29 (10 terms, d = 3), the sum equals (10/2)(2 + 29) = 5 x 31 = 155.

Sum of a Geometric Sequence

The sum of the first n terms of a geometric sequence, when r does not equal 1, equals a_1 times (1 - r^n), divided by (1 - r): S_n = a_1(1 - r^n) / (1 - r). The r = 1 case needs no formula, because every term is identical and the sum equals n times a_1. For the sequence 3, 6, 12, 24, 48 (5 terms, r = 2), the sum equals 3(1 - 2^5) / (1 - 2) = 3(-31) / (-1) = 93. Track the signs of the numerator and the denominator together, since both turn negative whenever r is greater than 1.

Sample Problem Walkthroughs

Arithmetic Sequence Problem

An arithmetic sequence has a 3rd term of 11 and a 7th term of 27. Find the 10th term. The two known terms sit four positions apart, so subtracting one equation from the other isolates d. Work through the steps below in order.

1. Write both terms with the nth-term formula: a_1 + 2d = 11, and a_1 + 6d = 27.
2. Subtract the first equation from the second: 4d = 16, so d = 4.
3. Substitute d back into the first equation: a_1 + 2(4) = 11, so a_1 = 3.
4. Compute the 10th term: a_10 = a_1 + 9d = 3 + 9(4) = 39.

Geometric Sequence Problem

A geometric sequence starts at 5, and its 4th term equals 135. Find the common ratio and the 6th term. The 4th term sits three steps from the first, so the exponent on r is 3. Work through the steps below in order.

1. Write the 4th term with the nth-term formula: 135 = 5 x r^3.
2. Divide both sides by 5: r^3 = 27, so r = 3.
3. Compute the 6th term: a_6 = a_1 x r^5 = 5 x 243 = 1215.

Recursive Sequence Problem

A sequence is defined by a_1 = 4 and a_n = 2 x a_(n-1) - 3 for n greater than 1. Find a_5. The rule doubles the previous term and subtracts 3, so four steps take you from a_1 to a_5. Work through the steps below in order.

1. a_2 = 2(4) - 3 = 5.
2. a_3 = 2(5) - 3 = 7.
3. a_4 = 2(7) - 3 = 11.
4. a_5 = 2(11) - 3 = 19.

Sequence problems reward a steady process over a shortcut you have not verified. Work through untimed reps first, check every step, then move to timed sets once your accuracy holds. We built free Sequences drill sets with instant feedback and a full explanation for every question, no signup required, alongside the rest of the Quantitative Reasoning topics on our GMAT practice page. Pair your practice with our GMAT score calculator to track a projected score as your accuracy improves. For the full Quantitative Reasoning topic breakdown, see our GMAT Quantitative Reasoning guide, and for the neighboring Counting and Probability topics, see our GMAT Statistics, Counting and Probability guide.

Sources

• GMAC: GMAT Exam Content
• GMAC: GMAT Exam Structure