Number Series and Coding: Test the Rule Against Every Term

Look for structure systematically

Check consecutive differences, second differences, ratios, alternating subsequences and familiar squares or cubes. Prefer a simple rule that fits every supplied term. A finite sequence can fit many mathematical rules; exam options and context usually identify the intended conventional pattern.

Difference pattern

Series: 3, 8, 15, 24, 35. Differences are 5, 7, 9, 11, so the next intended difference is 13 and the next term 48. Equivalently, terms follow n²+2n for n=1,2,3,4,5. Verifying the first term prevents an off-by-one mistake.

Alternating pattern

Series: 2, 10, 4, 20, 6, 30, 8, ?. Odd positions form 2,4,6,8; even positions form 10,20,30,40. The missing term is 40. A single difference sequence is unnecessarily complicated here.

Letter coding

For a stated shift of +2, A→C, B→D and Y→A with wraparound. CAT becomes ECV. Check each letter independently. A code example does not automatically prove that the same shift applies in every position unless the problem establishes that rule.

Position values

Using A=1 through Z=26, the sum for DOG is 4+15+7=26. If a question instead uses reverse positions, A=26 and Z=1. Write the convention before calculating.

Practice

1. 5,10,20,40,? → 80 by doubling.
2. 1,4,9,16,? → 25 as consecutive squares.
3. 7,11,17,25,35,? → 47; differences 4,6,8,10,12.
4. Under +1 coding, TRAIN → USBJO.
5. Under reverse alphabet substitution A↔Z, B↔Y, CAT → XZG.

Error diagnosis

If a rule fits only the last two terms, it is not adequately checked. If two plausible rules remain, use the options and all stated constraints rather than inventing an elaborate formula. For wrong-term questions, test whether correcting one term restores a consistent simple pattern.

Timed drill

Solve five short series, then write the rule beside every answer. Score the explanation as well as the result. Memorising isolated sequences does little to improve pattern recognition; comparing families of rules is more useful. These examples are original skills practice, not claims about exact future exam questions.