4.2 Number Sequences, Rule Discovery & Checking Alternatives

Number sequences, rule discovery and checking alternatives

Prefer the simplest complete rule

Any finite list can be fitted by many complicated formulas. Aptitude questions normally intend a simple, repeatable relationship visible across the series. A valid rule must explain every supplied transition and predict the missing term without special exceptions.

Start by writing first differences beneath the terms. For:

7, 11, 15, 19, ...

the differences are +4, +4, +4, so the next term is 23.

For:

3, 6, 12, 24, ...

the ratios are ×2, ×2, ×2, so the next term is 48.

Do not stop after finding a relationship between only the last two terms. Check it from the beginning.

Changing differences

Differences may themselves form a pattern:

4, 7, 11, 16, 22, ...

The increases are +3, +4, +5, +6, so the next increase is +7 and the next term is 29.

If first differences change by a constant amount, the second differences are constant. For square numbers:

1, 4, 9, 16, 25

first differences are 3, 5, 7, 9 and second differences are 2. Recognising n² is faster, but the difference method still verifies the structure.

Triangular numbers add successive integers:

1, 3, 6, 10, 15, ... adds 2, 3, 4, 5, then 6 to make 21.

Alternating operations

A sequence may alternate rules:

5, 10, 8, 16, 14, 28, ...

The pattern is ×2, −2, ×2, −2, ×2, so the next term is 26.

Write the operation over each arrow. If the series looks irregular, split odd and even positions:

2, 20, 4, 18, 6, 16, ...

Odd positions are 2, 4, 6; even positions are 20, 18, 16. The next odd-position term is 8.

Interleaving is common because it hides two simple sequences inside one list.

Position-based patterns

Terms may depend on their position n:

• n² gives 1, 4, 9, 16;
• n³ gives 1, 8, 27, 64;
• 2n + 1 gives 3, 5, 7, 9;
• powers of 2 give 1, 2, 4, 8, 16.

Check whether the starting index is clearly implied. A list beginning with 0 may shift a familiar formula.

Distinguish a recursive rule, which builds each term from the previous term, from a position rule, which calculates a term directly from its index. Either may describe the same list, but the question may ask specifically for one form. In a recursive sequence, verify the starting value as well as the operation; “double and add one” produces different series from different starts.

Combined multiply-and-adjust rules

For:

2, 5, 11, 23, 47, ...

each term is previous ×2 +1. The next is 95. Confirm:

2×2+1=5; 5×2+1=11; 11×2+1=23; 23×2+1=47.

Another series might use ×2, +1, ×2, +1 as alternating operations rather than applying both each time. Arrow notation distinguishes them.

Difference tables

For a harder polynomial-looking series, arrange levels:

Terms: 2, 7, 16, 29, 46

First differences: 5, 9, 13, 17

Second differences: 4, 4, 4

Continue the constant second difference: next first difference = 21, next term = 67.

Use this only after simpler additive, multiplicative, and alternating rules fail. Difference tables are systematic but can be slower.

Ratios and near-ratios

If numbers grow rapidly, inspect division. The sequence 81, 27, 9, 3 divides by 3. When ratios are not exact, test a combined rule such as ×2 −1.

Be cautious with decimal sequences. Convert to fractions or scale all terms by a power of ten if that makes the rule clearer.

Missing terms inside a sequence

For 4, 9, ?, 25, 36, recognise consecutive squares 2², 3², 4², 5², 6², so the missing term is 16. Verify both neighbours. An answer that fits the left transition but not the right is insufficient.

Use options intelligently

If several rules seem possible, substitute each option and see whether it maintains a simple pattern. Options are evidence, but avoid reverse-engineering a rule so elaborate that it exists only to select one choice.

Rank hypotheses:

1. constant difference or ratio;
2. simple alternating operations;
3. odd/even interleaving;
4. familiar position patterns;
5. changing differences or combined operations.

Avoid common traps

• Applying a rule only to the last pair.
• Confusing addition of a percentage with multiplication by a factor.
• Treating a visually familiar set as squares without checking every term.
• Forgetting that negative numbers can continue a difference pattern.
• Selecting the next difference instead of the next term.

State the rule in words and calculate one predicted term beyond the answer when possible. A rule that continues coherently is more credible than a one-step coincidence.