3.1 Number Series

On the NMAT, Inductive Reasoning is one of four Part 1 mental-ability subtests. CEM allocates 30 items and recommends roughly 35 minutes. Series items—number, letter, and figural—measure your ability to perceive relationships and infer a general rule when the full rule is never stated. You will not be told “this is arithmetic” or “this is geometric.” You must extract the rule from the given terms.

Number series are the most familiar type for most test-takers, but they are also where careless arithmetic and pattern-tunnel-vision cost the most points. This section builds a method first, then walks through fully worked examples with wrong-distractor analysis, then closes with a timed, no-calculator practice approach.

What a Number Series Item Asks

A typical stem presents a sequence with a blank at the end (or sometimes in the middle):

3, 7, 15, 31, __

Options are usually four numbers. Only one option continues the true underlying rule. Distractors often continue a plausible but incomplete rule (for example, only looking at first differences, or adding a constant that fits early terms but not all).

Your job: infer the rule that fits every given term, then apply it once more.

Core Pattern Families

1. Arithmetic series (constant difference)

Each term increases (or decreases) by the same amount d.

Example: 4, 9, 14, 19, 24 → d = +5 → next = 29.

2. Geometric series (constant ratio)

Each term multiplies by the same factor r.

Example: 2, 6, 18, 54 → r = ×3 → next = 162.

Watch for fractions and negatives: 81, 27, 9, 3 → r = ÷3 → next = 1.

3. Alternating series

Two interleaved rules—often one for odd positions and one for even positions.

Example: 2, 5, 4, 10, 8, 20, __

• Odd positions: 2, 4, 8 → ×2 each time → next odd term would be 16.
• Even positions: 5, 10, 20 → ×2 each time.

If the blank is the 7th term (odd position), answer is 16—not 40.

4. Two-step / compound rules

A single rule combines operations: “multiply by 2 then add 1,” “add 3 then multiply by 2,” and so on.

Example: 3, 7, 15, 31, 63

Check: 3×2+1=7, 7×2+1=15, 15×2+1=31, 31×2+1=63 → next = 127.

These are extremely common on entrance exams because first differences alone look irregular until you spot the compound step.

5. Digit-operation patterns

The rule acts on the digits of the previous term, not on the whole number as a pure arithmetic object.

Example: 12, 14, 18, 26, 38, __

Each term adds the product of its own digits: 12 + (1×2) = 14, 14 + (1×4) = 18, 18 + (1×8) = 26, 26 + (2×6) = 38, so the next term is 38 + (3×8) = 62.

The first differences here are +2, +4, +8, +12 — irregular enough to look random until you test the digits. Always try: sum of digits, product of digits, reverse digits, or “add the number formed by the digits.” Digit rules often produce jumps that look random if you only subtract consecutive terms.

6. Prime, square, cube, and other special sequences

Sometimes the terms are primes (2, 3, 5, 7, 11…), squares (1, 4, 9, 16, 25…), cubes, triangular numbers, or Fibonacci-like sums of previous terms. Sometimes the differences are primes or squares.

Example: 2, 3, 5, 7, 11, 13 → next prime = 17.

Example: 1, 4, 9, 16, 25 → next square = 36.

Example (differences are primes): 1, 3, 6, 11, 18 → differences +2, +3, +5, +7 → next difference +11 → next term 29.

The Difference-Table Method

When the pattern is not obvious in 5–10 seconds, write successive differences on scratch paper. This is your primary tool for arithmetic, two-step, and polynomial-like sequences.

How to build the table

Given: a, b, c, d, e

1. First differences: (b−a), (c−b), (d−c), (e−d)
2. Second differences: differences of those first differences
3. Stop when a row becomes constant (or clearly follows a simple rule like primes or ×2)

Worked example A — constant second difference

Series: 2, 5, 10, 17, 26, __

Second differences are constant (+2), so first differences continue +11, then next term = 26+11 = 37.

Distractor analysis: A student who only notices “+3 then bigger” might guess +11 from a partial pattern and still get 37, but a weaker distractor is 35 (if someone adds 9 again) or 36 (confusing with squares 25→36). The difference table prevents “add the last difference again” when second differences show the first differences themselves are growing.

Worked example B — geometric vs arithmetic trap

Series: 3, 6, 12, 24, __

First differences: +3, +6, +12 — not constant. Ratios: ×2, ×2, ×2. Next = 48.

Distractor analysis: Option 36 tempts students who switch to “add 12 again.” Option 30 tempts “add the primes 3+6+12 somehow.” The constant ratio is the rule that fits all terms.

Worked example C — two-step rule

Series: 5, 11, 23, 47, __

First differences: +6, +12, +24 → each difference doubles. That often signals ×2 then add something:

5×2+1=11, 11×2+1=23, 23×2+1=47 → next = 47×2+1 = 95.

Distractor analysis: Following only differences without checking the compound rule: next difference might be guessed as +48 → 47+48=95 (same answer here). But for 4, 9, 19, 39: differences +5, +10, +20; rule is ×2+1 → next 79, while naive “double last difference” gives 39+40=79. When they diverge, always verify the candidate rule by regenerating every given term.

Worked example D — alternating

Series: 8, 10, 7, 14, 6, 18, __

Separate positions:

• 1st, 3rd, 5th, 7th: 8, 7, 6, ? → decreasing by 1 → 5
• 2nd, 4th, 6th: 10, 14, 18 → +4 each

Blank is 7th term → 5.

Distractor analysis: Students who stay on a single chain might compute 6×3=18 and invent ×3 for the next term (54), or average neighboring terms. Interleaving is the trap that single-chain difference tables miss if you do not first check position parity.

Worked example E — primes in the differences

Series: 10, 12, 15, 20, 27, __

Differences: +2, +3, +5, +7 → primes. Next prime difference +11 → 27+11 = 38.

Distractor analysis: 35 (+8, continuing “+1 more than previous difference incorrectly as even steps”) or 36 (square attraction) are common wrong answers.

Digit-Operation and Special-Sequence Checklist

When first and second differences look ugly, run this 20-second checklist:

1. Ratios nearly constant? → geometric / near-geometric
2. Odd vs even positions separate cleanly? → alternating
3. ×k ± m regenerates all terms? → two-step
4. Sum/product of digits or reverse of previous? → digit rule
5. Known sequences: primes, squares (n²), cubes (n³), triangular (n(n+1)/2), Fibonacci (sum of two previous)
6. Differences themselves primes/squares/powers of 2?

Timed Practice Approach (No Calculator)

CEM’s recommended pace for Inductive Reasoning is about 70 seconds per item on average (35 min / 30 items). Number series should usually be faster than figural items—target 45–60 seconds when the pattern is standard.

On each item:

1. Scan (5–8 s): Look for ×2, obvious arithmetic, squares, primes.
2. Difference table (15–25 s): Write first (and if needed second) differences. Do not do it only in your head under time pressure.
3. Generate next term (5–10 s): Apply the rule; verify it against the last two given terms.
4. Match option (5 s): If your answer is not among the choices, your rule is wrong—recheck alternating and digit rules before recalculating arithmetic.
5. Skip trigger (at ~60–70 s): If nothing fits, mark and move on. Return only if time remains. Spending three minutes on one series is how strong students lose easier points later in the subtest.

Scratch-paper habits that save time:

• Write the series on one line; differences directly under the gaps between terms.
• Circle position numbers 1…n above terms when you suspect alternating rules.
• For geometric suspects, write the multiplier between terms (×2, ×3) instead of differences.
• Never rely on a calculator—none is provided, and mental arithmetic for two-digit ×2 and simple sums is expected.

Common Traps Summary

Master the difference table and the six pattern families above, and number-series items become a reliable score bank within the Inductive Reasoning subtest.