Inductive Reasoning

Infer patterns from examples and sequences

34 practice questionsPage updated 25 Sept 2026Reviewed by AISEA Editorial DeskPractice questions

Inductive reasoning moves from observed examples to a general rule or a likely prediction. In aptitude exams, the examples are usually terms in a number or letter sequence, a code transformation or a repeated arrangement. The task is to form a simple rule that explains every supplied term, test it against the choices and then extend it to the missing term. An inductive conclusion is supported by the evidence, not guaranteed by it, so a rule that fits only the last two terms is not enough.

Key formulas and rules

6 rules

Arithmetic Progression Formula: $T_n = a + (n - 1)d$ , where $a$ is the first term and $d = T_k - T_{k-1}$ is the common difference

Geometric Progression Formula: $T_n = a \cdot r^{n-1}$ , where $r = \frac{T_k}{T_{k-1}}$ is the common ratio

Difference Series Method: First differences $\Delta_1(n) = T_{n+1} - T_n$ ; second differences $\Delta_2(n) = \Delta_1(n+1) - \Delta_1(n)$ (constant $\Delta_2 \implies T_n = a n^2 + b n + c$ )

Interleaved Series Decomposition: For sequence $S$ , partition into odd subsequence $\langle T_1, T_3, T_5, \dots \rangle$ and even subsequence $\langle T_2, T_4, T_6, \dots \rangle$

Modular Alphabet Encoding: $\text{Pos}(A) = 1, \dots, \text{Pos}(Z) = 26$ ; cyclic shift by $k$ : $\text{NewPos} = ((p - 1 + k) \bmod 26) + 1$

Alphabet Numerical Product/Sum: $\operatorname{Value}(\text{Word}) = \sum_{i=1}^m \text{Pos}(c_i)$ or $\prod_{i=1}^m \text{Pos}(c_i)$

Key concepts

Induction is a tested hypothesis

A few observations can suggest a rule, but they do not prove that the rule is the only possible one. In a sequence question, treat each candidate rule as a hypothesis. Prefer the rule that explains all terms with the fewest arbitrary exceptions, then verify it on the missing position and the answer choices.

Start with first differences

Subtract each term from the next. Equal differences indicate an arithmetic sequence. If the differences are not equal, inspect whether they form a simpler sequence, such as consecutive odd numbers, consecutive primes or a steadily changing pattern. This is usually faster and safer than guessing a formula from the original terms.

Use second differences for curved growth

When first differences change regularly, calculate second differences by subtracting consecutive first differences. Constant second differences indicate a quadratic pattern. If second differences themselves change in a simple way, continue that smaller pattern and rebuild the next first difference before finding the next term.

Check ratios and changing multipliers

Divide a term by the previous term when the values grow or shrink multiplicatively. A sequence may use multipliers such as one-half, one, three-halves, two and five-halves rather than one constant ratio. Write the operation between every pair; the operation list is often the real sequence.

Split alternating or interleaved terms

If consecutive terms do not share one rule, separate positions 1, 3, 5 and so on from positions 2, 4, 6 and so on. Each subsequence may follow its own arithmetic, geometric or letter pattern. Recombine them only after both subsequences produce consistent results.

Decode letters numerically

Convert letters to positions when a code involves a shift, reversal, product or another arithmetic operation. State the direction of a shift and wrap Z back to A when needed. For a product code, calculate every letter position and multiply; do not infer the result from the word's length alone.

Use number properties as constraints

Some sequences are defined by primes, odd or even numbers, squares, cubes or divisibility. Identify the property first, then check the order or operation that links the observed values. A prime-related question may use prime differences or prime positions, so do not assume that the terms themselves must all be prime.

Reject overfitted patterns

Many different formulas can match a short list. Avoid a rule that needs a special exception for one term, changes operation without a visible cycle or ignores earlier values. Use answer choices as a verification tool, not as a substitute for a rule, and check the rule from the first term through the requested position.

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Worked examples

Example 1

Find the missing term in the sequence 3, 14, 39, 84, ?, 258.

Solution

Step 1: Number the positions from 1 to 6.
Step 2: The terms fit n³ + n² + n: for n = 1, the value is 3; for n = 2, it is 14; for n = 3, it is 39; and for n = 4, it is 84.
Step 3: At n = 5, the missing term is 5³ + 5² + 5 = 125 + 25 + 5 = 155.
Step 4: At n = 6, the rule gives 216 + 36 + 6 = 258, confirming the pattern.
Answer: 155.

Example 2

Find X in the sequence 132, 129, 124, 117, 106, 93, X.

Solution

Step 1: Find the successive decreases: 3, 5, 7, 11 and 13.
Step 2: These are consecutive prime numbers beginning with 3.
Step 3: The next prime decrease is 17.
Step 4: X = 93 − 17 = 76.
Answer: 76.

Example 3

Find X in the sequence 20, 10, 10, 15, 30, 75, X.

Solution

Step 1: Write the multiplier between each pair: one-half, one, three-halves, two and five-halves.
Step 2: The multipliers increase by one-half each time, so the next multiplier is three.
Step 3: X = 75 × 3 = 225.
Answer: 225.

Example 4

If ZERO is written as CHUR by shifting every letter three positions forward, how is PLAYER written? Use alphabet wraparound.

Solution

Step 1: Shift each letter three positions forward: P becomes S, L becomes O, A becomes D, Y becomes B, E becomes H and R becomes U.
Step 2: Y wraps around because the alphabet has 26 positions.
Answer: SODBHU.

Example 5

A code gives ABCD as 24 and EFGH as 1,680. If the rule is the product of the alphabet positions, what is the code for IJKL?

Solution

Step 1: A, B, C and D have positions 1, 2, 3 and 4, whose product is 24.
Step 2: E, F, G and H have positions 5, 6, 7 and 8, whose product is 1,680. This confirms the rule.
Step 3: I, J, K and L have positions 9, 10, 11 and 12.
Step 4: Product = 9 × 10 × 11 × 12 = 11,880.
Answer: 11,880.

Example 6

The sequence 2, 5, 4, 10, 8, 20, ? is interleaved. Find the next term.

Solution

Step 1: Odd-position terms are 2, 4, 8, so each term doubles.
Step 2: Even-position terms are 5, 10, 20, which also double.
Step 3: The seventh term continues the odd-position subsequence: 8 × 2 = 16.
Answer: 16.

Example 7

The first terms of a sequence are 4, 9, 16, 25. Two possible hypotheses are ‘add consecutive odd numbers’ and ‘square the position plus three’. Which hypothesis should be accepted for predicting the fifth term?

Solution

Step 1: The terms are 2², 3², 4² and 5², so the simplest supported rule is the square sequence.
Step 2: The same pattern can be described through differences: add 5, then 7, then 9.
Step 3: The fifth term is 6² = 36.
Step 4: The two descriptions agree here, but the rule should still be checked against every supplied term rather than selected from one difference alone.
Answer: 36, with the square-of-consecutive-integers rule.

Computational ThinkingOpenStax · Checked 1 Sep 2026Supports: pattern recognition, generalization and moving from observations to a tentative hypothesis

Types of InferencesOpenStax · Checked 1 Sep 2026Supports: inductive reasoning as probable rather than guaranteed inference from experience

Informal FallaciesOpenStax · Checked 1 Sep 2026Supports: weak induction and the risk of drawing broad conclusions from insufficient or narrow evidence

Inductive Reasoning

Key formulas and rules

Key concepts

Induction is a tested hypothesis

Start with first differences

Use second differences for curved growth

Check ratios and changing multipliers

Split alternating or interleaved terms

Decode letters numerically

Use number properties as constraints

Reject overfitted patterns

Worked examples

Example 1

Example 2

Example 3

Example 4

Example 5

Example 6

Example 7

Representative solved questions

Question 1

Question 2

Question 3

Common mistakes and useful habits

Sources and review notes

Ready to test your understanding?

Key formulas and rules

Key concepts

Induction is a tested hypothesis

Start with first differences

Use second differences for curved growth

Check ratios and changing multipliers

Split alternating or interleaved terms

Decode letters numerically

Use number properties as constraints

Reject overfitted patterns

Worked examples

Example 1

Example 2

Example 3

Example 4

Example 5

Example 6

Example 7

Representative solved questions

Question 1

Question 2

Question 3

Common mistakes and useful habits

Sources and review notes

Ready to test your understanding?

Keep building the same skill family.