Sequences: Write Out Five Terms Before You Look for a Formula

By EMPOWERgmat on July 21, 2026

I watch the same scene every week. A Test Taker hits a sequence question, sees a letter with a tiny n tucked beside it, and freezes. Their eyes drift up and to the left. They're trying to remember a formula from high school. Something with an n minus 1 in it. Was it divided by 2? Multiplied by the first term? Ninety seconds later the scratch paper is still blank, and they guess.

That is NOT what you should do here. The fix is almost insultingly simple. Write out five terms. Then look at them. On most GMAT sequence questions, the pattern jumps off the page by term four or five, and the question is basically over.

What a sequence question really is

A sequence is just a list of numbers in a set order. Each number in the list is called a term. The GMAT often writes the term in spot n as aₙ, so a₁ is the first term, a₂ is the second, and a₁₀₀ is the hundredth.

The question gives you a rule for building the terms. Sometimes the rule uses the term's position, like aₙ = 3n + 1. Sometimes each term is built from the terms before it, like "each term is double the one before." That second kind is called recursive, which just means "built from what came before."

Here's the part people miss. The test writers KNOW you can't write out 100 terms in two minutes. That's exactly why they ask for the 100th term, or the sum of the first 20. They're daring you to go hunting for a formula. But they built the question so the pattern shows up early. The formula is the long way. The list is the short way.

And time matters. The Quantitative Reasoning section gives you 21 Problem Solving questions in 45 minutes. That averages out to a little over 2 minutes each. You do not have 90 seconds to spend staring at the ceiling trying to recall something.

So decide what you want MORE: to feel like a mathematician, or to get the point?

Try one

Try it

In a certain sequence, each term is defined by aₙ = 1/n − 1/(n + 1). What is the sum of the first 20 terms of the sequence?

19/20
20/21
21/22
1
21/20

Show the answer

Step 1: Write five terms on paper. Don't simplify them yet.

a₁ = 1 − 1/2
a₂ = 1/2 − 1/3
a₃ = 1/3 − 1/4
a₄ = 1/4 − 1/5
a₅ = 1/5 − 1/6

Step 2: Add them and watch what happens. The −1/2 from the first term cancels the +1/2 from the second. The −1/3 cancels the +1/3. Every middle piece wipes out. All that's left is the first piece and the last piece: 1 − 1/6 = 5/6.

Step 3: Spot the pattern. The sum of the first 5 terms is 1 − 1/6. Check it smaller: the first term alone is 1 − 1/2 = 1/2, and the first 2 terms add to 1 − 1/3 = 2/3. So the sum of the first n terms is always 1 minus 1 over (n + 1).

Step 4: Apply it. The 20th term is 1/20 − 1/21, so the leftover piece is −1/21. The sum is 1 − 1/21 = 20/21. The answer is (B).

Quick check: every term is positive, and the sum is 1 minus something, so it must be less than 1. That kills (D) and (E) before you do any real work.

The trap: being off by one. People remember "it cancels down to 1 minus the last fraction" and grab 1 − 1/20 = 19/20, which is (A). The last term ends in 1/21, not 1/20. Your written list protects you here, because you can SEE that five terms leave 1/6 behind, not 1/5. The other trap is turning each term into a decimal and adding 20 of them. That works eventually. So does walking to the test center.

A line of dominoes where each domino's right half matches the next domino's left half, the matching halves fade away, and only the two outer ends stay solid. — Write the terms out and the middle cancels itself.

Five terms is the minimum, not the maximum

Sometimes five terms aren't enough to see it. That's fine. Five is where you START looking. You keep writing until the pattern repeats or the rule is obvious. This next one needs a few more.

Try it

In a sequence, a₁ = 2 and a₂ = 5. Each term after the second equals the term just before it minus the term two spots before it. What is a₁₀₀?

−5
−3
−2
3
5

Show the answer

Step 1: Write the terms in a row, with the position above each one.

Position	1	2	3	4	5	6	7	8
Term	2	5	3	−2	−5	−3	2	5

Here's the work for each one. a₃ = 5 − 2 = 3. a₄ = 3 − 5 = −2. a₅ = −2 − 3 = −5. a₆ = −5 − (−2) = −3. a₇ = −3 − (−5) = 2. a₈ = 2 − (−3) = 5.

Step 2: Spot the pattern. Terms 7 and 8 are 2 and 5, the same as terms 1 and 2. Since every term is built from the two before it, the whole list now repeats. The cycle is 6 terms long: 2, 5, 3, −2, −5, −3.

Step 3: Find where 100 lands. 100 ÷ 6 = 16 with a remainder of 4. So 16 full cycles take you through term 96, and term 100 is the 4th spot of the next cycle. The 4th term of the cycle is −2. The answer is (C).

Check it a second way: 96 is a multiple of 6, so a₉₆ is the last spot of a cycle, which is −3. Count forward: a₉₇ = 2, a₉₈ = 5, a₉₉ = 3, a₁₀₀ = −2. Same answer.

The trap: stopping too early. At five terms you have 2, 5, 3, −2, −5, and it looks like nonsense. Keep going. The other trap is the remainder. If you think term 100 sits at the END of a cycle, you pick −3, which is (B). If you miscount by one spot, you land on −5, which is (A). Both wrong answers are sitting there waiting for you. That's on purpose.

That remainder step should feel familiar. It's the same move you use when small numbers reveal a remainder pattern. Find the cycle, divide by its length, read off the spot.

A row of colored tiles where the same six-color pattern repeats again and again, with a curved arrow looping from the end of one set back to its start. — Once a pattern repeats, you only need one lap of it.

The mistakes I see every week

Hunting for a formula you half remember. A formula you're 70% sure of is a formula that's 30% likely to cost you the point.
INSTEAD: Write terms first, every time. If a formula shows up from the list, great. If not, the list itself answers the question.
Doing the terms in your head. Some Test Takers think they're 'smart enough' to track five negative numbers mentally. THAT is how a −5 turns into a 5.
INSTEAD: Write every term on paper, in a row, with its position number above it. It takes seconds and it removes the silly mistake.
Stopping at three terms. Three terms can fit a dozen patterns. You'll see one, trust it, and be wrong.
INSTEAD: Write at least five. Keep going until something repeats or the rule is impossible to miss.
Getting the position wrong by one. This is the single most common way to lose a sequence question you actually solved.
INSTEAD: Test your pattern on a term you already wrote. If your rule says the first 5 terms add to 5/6, does your list agree? If not, fix the rule before you use it.
Treating sequences as 'hard' questions to skip. They look scary. Most of them are gettable, and gettable points are where your score lives.
INSTEAD: Give them 30 seconds of writing before you decide anything. You'll be surprised how often the list does the work.

Why this works on more than sequences

Writing out terms is just one version of a bigger habit. Make the problem concrete, put it on paper, and let the pattern show itself. It's the same reason you draw the grid on overlapping sets and picture the numbers on median and range questions. Quant is not a math test. It's a thinking test that uses math as the gauge, and the Test Takers who score highest are the ones who stop acting like mathematicians and start acting like strategists.

Think of a tennis player learning a new court. She doesn't read about the bounce. She hits a few balls and watches. Five terms is you hitting a few balls.

I'll bet the last sequence question you missed, you missed with a nearly empty page. Go check. I'll wait.

Your move today

Pull three sequence questions from the official practice materials you already have, including at least one you got wrong before. For each one, write five terms on paper before you think about any formula. Time yourself. Then compare that time to how long the question took you the first time. If writing the list was faster AND you got it right, you have your answer about how to handle the next one on Test Day.