Concepts · Number series
Stop staring at the numbers. Write what happens between them.
Kabir has a number series in front of him:
2, 5, 11, 20, 32, ?
Times two? No. Squares? No. He looks at the numbers and waits for the pattern to jump out at him. It does not jump. After a minute he picks the option that looks about the right size.
At home, his little sister’s height is marked in pencil on the door frame, one line every birthday: 87, 96, 104, 111. She wants to know how tall she will be next year.
“You grew nine, then eight, then seven,” Kabir tells her. “So about six more. 117.”
Then he hears what he just did. He never looked at the heights. He looked at how much she grew between the lines, and at how the growing was changing. He takes the mock paper back out.
Write the gaps under the series. If they are not all the same, write the gaps of the gaps. Keep going until a row of at least three numbers is all the same. Then extend that row by one and add your way back up.
No idea had to arrive. The table did the noticing.
Sometimes the gaps of the gaps are not all the same either. 1, 2, 6, 15, 31, 56, ? has gaps 1, 4, 9, 16, 25, then 3, 5, 7, 9, then 2, 2, 2. The bottom row extends to 2, so the middle row goes on to 11, the gaps to 25 + 11 = 36, and the series to 56 + 36 = 92. Same method, one floor deeper.
And if three rows down nothing settles, the series does not grow in a way the table can catch. Stop, and look elsewhere: for landmarks in the terms (squares, cubes), for a multiplying rule, or in the gaps themselves (are the gaps primes, or doubling?): Landmark numbers.
A series built by adding keeps its rule between the terms, not in them. Waiting for the pattern to jump out can take a minute and end in a guess. The table takes the same numbers and shows the rule in a few lines of subtraction, and when it finds nothing, it tells you to look for a landmark instead.
1. Find the next term in the series:
3, 5, 9, 15, 23, ?
Write the gaps underneath. Are they the same? If not, write the gaps of the gaps.
33
The gaps are 2, 4, 6, 8. Their gaps are 2, 2, 2: all the same.
The next gap is 8 + 2 = 10, so the next term is 23 + 10 = 33.
2. Find the next term in the series:
2, 6, 12, 20, 30, ?
The gaps grow by the same amount each time.
42
The gaps are 4, 6, 8, 10, growing by 2.
The next gap is 12, so the next term is 30 + 12 = 42.
3. Find the next term in the series:
1, 3, 8, 16, 27, ?
The gaps of the gaps are all the same. Which number are they?
41
The gaps are 2, 5, 8, 11. Their gaps are 3, 3, 3.
The next gap is 11 + 3 = 14, so the next term is 27 + 14 = 41.
A week later: 3, 7, 13, 21, 31, ? Kabir does not wait for anything. Gaps 4, 6, 8, 10; their gaps 2, 2, 2. The next gap is 12. He writes 43.
Number series is one of the app’s fifteen topics. Every question there is answered before its solution shows.
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