Sequences and Patterns: Find the CYCLE

Say you're nine minutes into your first Quant section. You've been moving well. Then a question shows you two numbers, a rule for making the next number, and asks for the 100th one. Your stomach drops. You start hunting your memory for a formula. You find nothing. You burn three minutes, guess, and move on feeling rattled.

Here's my verdict: that question is one of the EASIEST on the test. It only looks hard because of the big number. The GRE is not asking you to do 100 steps of math. It's asking whether you'll do six steps on paper and then notice something.

What the question is really testing

A sequence is just a list of numbers in a set order. Each number in the list is called a term. The first term, the second term, and so on. The question usually hands you a rule for building the next term out of the ones before it.

When the terms start to repeat in the same order, that repeating block is called a cycle. Once you know the cycle, the 100th term is not a math problem anymore. It's a counting problem.

Now think about the clock. As of today, the GRE gives you two Quant sections: 12 questions in 21 minutes, then 15 questions in 26 minutes. That's under two minutes per question on average. You do NOT have time to invent a formula from scratch. You DO have time to write eight numbers on your scratch paper. You can see the full layout on ETS's Quant section page.

Let's do one.

The one step that trips people: the remainder

Step 4 above is really a remainder question. A remainder is what's left over after you divide. Divide the position you want by the length of the cycle. The remainder tells you where you land inside the cycle. If you want a refresher, we break it down in Remainders Made SIMPLE.

Here's the part people blow. A remainder of 0 does NOT mean the first term. It means the LAST term of the cycle, because you just finished a full lap. Don't trust me on that. Test a small value. In a cycle of 6, which term is term 6? The last one. And 6 divided by 6 leaves a remainder of 0. Done. You just proved it yourself in five seconds.

Same idea, different costume: units digits

The GRE loves to hide a cycle inside a giant power. The units digit is the ones digit, the last digit of a number. You can't work out 3 to the 47th power by hand. You don't need to. The units digit of a power only depends on the units digit of the number you're multiplying, so those last digits cycle fast.

This next one is a Quantitative Comparison. That's a GRE question type where you compare two quantities and pick which is bigger, whether they're equal, or whether you can't tell.

Notice what you did in both questions. No formula. No genius. You wrote short lists, looked at them, and counted. That's the same move behind thinking of absolute value as DISTANCE instead of a pile of rules. Find the simple idea underneath and the scary question shrinks.

5 mistakes I see every week

1. Hunting for a formula. You stare at the rule and try to remember a trick from school. The clock runs. INSTEAD: start writing terms the second you finish reading. The pattern shows up on the page, not in your memory.
2. Stopping too early. You see one number repeat and call it a cycle. One repeat is not a cycle. INSTEAD: keep going until the opening run of terms shows up again in the same order. If the rule uses the two terms before, you need TWO matching terms in a row.
3. The remainder zero slip. You get a remainder of 0 and grab the first term. INSTEAD: remainder 0 means the last term of the cycle. If you ever doubt it, test a tiny position you can check by hand.
4. Assuming the cycle starts at term 1. Some sequences wander for a few terms before they settle into a loop. INSTEAD: mark exactly where the repeat starts. Count off the lead in terms first, then count laps.
5. Doing it in your head. I'll bet the people who say they're "good at mental math" are the ones landing one spot off. That's pride, and it costs points. INSTEAD: write position and value side by side on your scratch paper, every single time. It takes seconds and kills the off by one mistake.

Your move today

Go back to the first example. Using the cycle you already wrote, find the sum of the first 100 terms. Hint: add up one full cycle first and look hard at what you get. Then use the same counting you just learned. Do it on paper. When you're done, your answer should be a single digit. If it isn't, write the list again and recount.

Then pull three sequence or units digit questions from your practice material. Before you touch anything else, write out the first eight terms. Every time. That's the whole habit, and it starts tonight.