Number Sequences & Function Machines

OC sequences go beyond adding: 2, 6, 18, 54 TRIPLES each time. Spot multiply-rules by the jumps — if the gaps themselves grow (4, 12, 36…), an add-rule can't explain it. A typical OC question: find the difference between the 3rd and 5th terms of 2, 6, 18, 54, … Extend: 5th = 162. Then the second step: 162 − 18 = 144. Extend first, THEN answer what was actually asked.

OC rules can have two parts: 'each term is 5 more than three times the one before'. Forward from 2: 11, 38, 119 — both parts, every step. The twist runs it BACKWARDS: what term comes directly before 41? Undo in reverse order: undo the +5 (41 − 5 = 36), then undo the ×3 (36 ÷ 3 = 12). Test questions often go one step further: add the digits of 12 → 3. Three-step questions are normal here.

A function machine eats a number, applies its rule, and spits out the result: rule '×3 then +2' turns 5 into 17. OC gives you in/out pairs and asks for the rule or a missing output. Detective trick: check how much OUT grows when IN grows by 1 — if out jumps by 3 each time, a ×3 lives inside; then one pair reveals the add-on (in 2 → out 7: 2×3=6, so +1).

Repeating patterns hide a block. OC style: a bracelet repeats '3 green then 4 white'; it has 20 white beads — how many green? Think in BLOCKS of 7: 20 white ÷ 4 per block = 5 blocks, so green = 5 × 3 = 15. Position questions use the same block: 'what colour is the 28th bead?' — 28 ÷ 7 = 4 exact blocks, so bead 28 ENDS a block: white. Leftovers tell you how far into the next block you are.

Three traps guard OC sequence questions. One: term NUMBER vs VALUE — the '5th term' is the fifth number in the list, not the number 5. Two: 'is 40 in the pattern?' — extend it honestly; patterns can jump straight over a number (…35, 41). Three: the after-question — you extend perfectly, then miss that they wanted a difference or a digit-sum of your result. Cure for all three: after your working, RE-READ the question and answer exactly that. Quiz and exercises now!