7.3 Number & Letter Sequences and Series

Reading the Pattern

Sequence items hand you a short list and ask for the missing or next term. The Professional level mixes arithmetic, geometric, layered-difference, alternating, and letter series, so guessing is unreliable. A systematic method wins every time.

The Four-Step Pattern-Spotting Method

1. Differences. Write the gap between each pair of consecutive terms. A constant difference means the sequence is arithmetic.
2. Ratios. If the differences are not constant, divide each term by the one before it. A constant ratio means the sequence is geometric.
3. Go deeper. If neither is constant, look at the pattern OF the differences (the second differences), or test whether two separate series are interleaved.
4. Verify. Confirm your rule reproduces every given term before selecting an answer. Two terms alone can fit many rules.

Arithmetic Sequences

A constant difference d is added at each step, and the nth term = first term + (n - 1)d.

Worked Example 1. 4, 9, 14, 19, 24, ? The differences are all +5, so the next term is 29. To leap far ahead, the 20th term = 4 + 19 x 5 = 4 + 95 = 99.

Geometric Sequences

A constant ratio r is multiplied at each step.

Worked Example 2 (bank item). 3, 6, 12, 24, 48, ? Each term is x2 (6/3 = 2, 12/6 = 2), so the next is 96.

Worked Example 3 (dividing series, bank item). 81, 27, 9, 3, ? Each term is divided by 3 (ratio 1/3), so the next is 1. A shrinking sequence is still geometric.

Layered or Growing-Difference Series

Here the differences themselves follow a pattern, so the first pass is not constant but the second pass is.

Worked Example 4 (bank item). 2, 5, 10, 17, 26, ? Differences are 3, 5, 7, 9, which are consecutive odd numbers, so the next difference is 11 and the next term is 26 + 11 = 37. These terms equal n x n + 1 (that is, 1+1, 4+1, 9+1, 16+1, 25+1).

Worked Example 5 (shrinking gaps, bank item). 100, 96, 88, 76, 60, ? Differences are -4, -8, -12, -16, growing by -4 each time, so the next difference is -20 and the term is 60 - 20 = 40.

Two-Step and Fibonacci-Style Series

When both differences and ratios refuse to settle, the rule often combines two operations or builds each term from earlier ones.

Worked Example 5b (multiply then add). 2, 5, 11, 23, ? Test x2 then +1: 2 x 2 + 1 = 5, 5 x 2 + 1 = 11, 11 x 2 + 1 = 23, so the next term is 23 x 2 + 1 = 47. A telltale sign is that each term is a little more than double the one before it.

Worked Example 5c (Fibonacci-style). 1, 1, 2, 3, 5, 8, ? Each term is the SUM of the two before it, so the next is 5 + 8 = 13. The giveaway is that every term roughly equals the previous two combined; this pattern never shows a constant difference or ratio, so do not waste time hunting for one.

Alternating (Interleaved) Series

Two separate series take turns filling the positions.

Worked Example 6 (bank item). 1, 4, 2, 5, 3, 6, ?, ? The odd positions read 1, 2, 3, so the next is 4; the even positions read 4, 5, 6, so the next is 7. The next two terms are 4, 7. Splitting by position exposes two simple +1 series hiding inside one list, which is why a single rule refuses to fit the whole thing.

Letter and Alphanumeric Series

Convert letters to their position numbers (A = 1, B = 2, up to Z = 26), solve the number pattern, then convert back. Memorize anchor letters to count fast: E = 5, J = 10, O = 15, T = 20, Y = 25.

Worked Example 7 (bank item). A, C, F, J, O, ? Positions are 1, 3, 6, 10, 15, with differences +2, +3, +4, +5. The next difference is +6, giving position 21 = U.

Worked Example 8 (skip pattern). B, D, F, H, ? These are the even letters, +2 each, so the next is J.

Worked Example 9 (alphanumeric). A1, C2, E3, G4, ? The letters skip one each time (A, C, E, G, then I) and the numbers count 1, 2, 3, 4, then 5, giving I5.

Worked Example 10 (reverse). Z, X, V, T, ? Positions 26, 24, 22, 20 drop by 2, so the next is position 18 = R. If a computed position exceeds 26, subtract 26 to wrap around (position 28 becomes B).

Common Mistakes

Three errors sink most sequence items: stopping at the first rule that fits only two terms instead of verifying against the whole list; miscounting letter positions, especially the off-by-one at A = 1; and forgetting the alternating possibility when a single rule stubbornly fails. When a list refuses every simple rule, split it into odd and even positions before giving up.