Have a checking order, not a hunch
Students stare at a series waiting for the pattern to appear. Under exam pressure it often does not. Replace staring with a fixed order of checks. You will usually land the pattern inside two of them.
Step 1. Write the differences. Under each gap, write next minus previous. If the differences are constant, it is arithmetic. If the differences themselves form a simple pattern (2, 4, 6, 8 or 3, 5, 7, 9), you are done.
Step 2. Check ratios. Divide each term by the one before. A constant ratio means a geometric series.
Step 3. Check squares and cubes. Keep these in your head: squares to 20 (1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400) and cubes to 10 (1, 8, 27, 64, 125, 216, 343, 512, 729, 1000). Also look for "square plus 1" or "cube minus 2" — a term that sits one or two away from a perfect square is a strong hint.
Step 4. Check the mixed rule *a + b. Look for doubling with a small correction: *2 + 1, *3 - 2, *2 - 3.
Step 5. Check alternate terms. Take the 1st, 3rd, 5th as one series and the 2nd, 4th, 6th as another. Two interleaved series look like nonsense until you split them.
Step 6. Check primes. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37.
When more than one rule fits the terms given, choose the one that is simplest and uses every term. If a rule works for four terms out of five, it is the wrong rule. Do not fix it with an exception.
Letter series needs two tools
Tool 1: positions. A = 1 to Z = 26. Learn the mnemonic EJOTY: E = 5, J = 10, O = 15, T = 20, Y = 25. From any of these you count a step or two rather than counting from A.
Tool 2: the opposite letter. A letter and its opposite always add to
- A and Z, B and Y, C and X, D and W, E and V. The mnemonic is
AZ BY CX DW EV. So the opposite of M (13) is N (14), since 13 + 14 = 27.
With those, treat letters as numbers. Convert, find the pattern in numbers, convert back.
Worked problem 1
Find the next term: 3, 6, 11, 18, 27, ?
Write the differences:
6 - 3 = 3
11 - 6 = 5
18 - 11 = 7
27 - 18 = 9
The differences are 3, 5, 7, 9 — consecutive odd numbers. The next difference is 11.
Answer = 27 + 11 = 38
A second reading of the same series: 3 = 1 + 2, 6 = 4 + 2, 11 = 9 + 2, 18 = 16 + 2, 27 = 25 + 2. So each term is a perfect square plus 2, and the next is 36 + 2 = 38. Two routes, one answer. That agreement is a good confidence check.
Worked problem 2
Find the next term: 5, 11, 23, 47, ?
Differences: 6, 12, 24. They are doubling, which points at a *a + b rule rather than a difference rule.
Test *2 + 1:
5 * 2 + 1 = 11 OK
11 * 2 + 1 = 23 OK
23 * 2 + 1 = 47 OK
Answer = 47 * 2 + 1 = 95
Worked problem 3
Find the next term: AZ, CX, EV, GT, ?
Split each pair. Handle the first letters and the second letters as two separate series.
First letters: A, C, E, G → positions 1, 3, 5, 7. Step of +2. Next is position 9 = I.
Second letters: Z, X, V, T → positions 26, 24, 22, 20. Step of -2. Next is position 18 = R.
Answer = IR
Notice the extra structure: A and Z are opposites, C and X are opposites, E and V are opposites. I and R should be too, and 9 + 18 = 27. Correct.
Always split a two-letter or three-letter series into columns before doing anything else. Almost every letter series in a placement paper is two simple series printed side by side.
Odd-one-out is the same skill
"Which does not belong: 16, 25, 36, 49, 64, 81, 100?" Find the rule that covers most of them — here, perfect squares — then find the one that breaks it. If they are all squares, look one level deeper: squares of even numbers against squares of odd numbers.
Practice set
Try these before the next lesson.
- 7, 14, 28, 56, ? (112)
- 2, 5, 10, 17, 26, ? (37; squares plus 1)
- 1, 4, 27, 256, ? (3125; n to the power n)
- BD, FH, JL, NP, ? (RT)
- 120, 99, 80, 63, 48, ? (35; squares minus 1, descending)