1.1 Pattern recognition & number/letter sequences

Reading a sequence like a cyber analyst

The DoD Cyber Test uses number and letter sequences because cyber work is fundamentally pattern work: spotting the one anomalous entry in a log, recognizing a repeating byte structure in captured traffic, or predicting the next value a system will generate. Each sequence item shows a short row of terms and asks for the next value or a missing middle value. Your job never changes — find the rule that turns one term into the next, confirm that the same rule holds across every gap in the row, then apply it one more time. Because the test is computer-adaptive and timed, a fast, mechanical procedure beats staring at the numbers and hoping the answer jumps out.

A four-step method that always works

1. Scan the entire row before writing anything. The rule sometimes only reveals itself at the third or fourth gap, so early guesses mislead.
2. Write the first differences: term2 minus term1, term3 minus term2, and so on. If every difference is identical, the sequence is arithmetic.
3. If the differences grow or shrink, test ratios: term2 divided by term1, term3 divided by term2. A constant ratio means the sequence is geometric.
4. If neither is constant, check the second differences, the sum of the two prior terms (Fibonacci-style), or an alternating pattern in which two separate streams are interleaved.

Arithmetic and geometric sequences

An arithmetic sequence adds a fixed number each step. Given 4, 9, 14, 19, ?, the first differences are +5, +5, +5, so the next term is 19 + 5 = 24. The general term is a1 + (n - 1)d, where d is the common difference. Arithmetic sequences can also decrease: 40, 33, 26, 19, ? subtracts 7 each time, giving 12. A geometric sequence multiplies by a fixed number instead. Given 3, 6, 12, 24, ?, each term doubles, so the next is 48. Geometric sequences can also shrink: 80, 40, 20, 10, ? multiplies by one-half, giving 5. Rule of thumb: if terms grow slowly and evenly, suspect arithmetic; if they explode or collapse, suspect a multiplier.

Fibonacci-style patterns and second differences

In a Fibonacci-style sequence each term is the sum of the two terms before it. Given 2, 3, 5, 8, 13, ?, add 8 + 13 = 21. Other sequences hide their rule in the second differences. Consider 2, 6, 12, 20, 30, ?: the first differences are 4, 6, 8, 10 (not constant), but the second differences are 2, 2, 2 (constant). The next first difference must therefore be 12, so the next term is 30 + 12 = 42.

Alternating (interleaved) sequences

Sometimes two independent patterns are woven together. In 5, 100, 10, 90, 15, 80, ?, the odd positions (5, 10, 15, ...) climb by +5 while the even positions (100, 90, 80) fall by -10. The question mark sits at position 7, an odd slot, so it continues the climbing stream: 15 + 5 = 20. Whenever no single rule fits the whole row, split it into the position 1-3-5 stream and the position 2-4-6 stream and analyze each separately.

Letter sequences: convert to numbers first

Map the alphabet to positions: A = 1, B = 2, on through Z = 26. A letter sequence then becomes a number sequence you already know how to solve. Given C, F, I, L, ?, the positions are 3, 6, 9, 12 — an arithmetic +3 pattern — so the next position is 15, which is O. Given B, D, G, K, ?, the positions are 2, 4, 7, 11 with differences +2, +3, +4; the next difference is +5, giving position 16 = P. If a computed position exceeds 26, wrap around by subtracting 26: position 27 becomes 1 = A.

Recognizing special number families

Some rows are not add-or-multiply patterns at all but recognizable families you can name on sight:

• Perfect squares: 1, 4, 9, 16, 25, 36 (each is n times n).
• Cubes: 1, 8, 27, 64, 125.
• Primes: 2, 3, 5, 7, 11, 13, 17.
• Triangular numbers: 1, 3, 6, 10, 15 (add 1, then 2, then 3, and so on).

A final small family is the repeating cycle: in 7, 2, 9, 7, 2, 9, ?, the block 7, 2, 9 simply repeats, so the next term restarts the cycle at 7.

Common traps to avoid

• Guessing from two terms. Two points fit infinitely many rules; verify across at least three gaps.
• Missing a decreasing direction — a sequence can subtract or divide, not only add or multiply.
• Reading two interleaved streams as one — the usual reason nothing seems to work.
• Letter wrap-around past Z, and confusing a letter's position with its value.
• Stopping at first differences when the real rule lives in the second differences.