Quantitative Aptitude · Number Series, Coding-Decoding and Odd Man Out

Number Series: Missing and Wrong Terms for CA Foundation

Updated 1 October 2026

A number series follows a hidden rule linking each term to the next. To find a missing or wrong term, check differences, ratios, squares, cubes and mixed rules like ×2 + 1. Test the rule on the first few terms, confirm it on the rest, then fill the gap or spot the term that breaks it.

Understand Number Series: Missing and Wrong Terms

A number series is a list of numbers built by one fixed rule. Your job is to find that rule. Once you know it, you can find a missing term or spot a wrong term (one that breaks the rule).

The rule is usually one of a few types. It may add or subtract a fixed number. It may multiply or divide by a fixed number. It may use squares or cubes, such as n² + 1 or n³ − 1. It may be mixed, such as ×2 + 1, or the differences themselves may form a pattern.

A good way to start is to look at the gaps between terms. If the gaps are equal, it is an arithmetic pattern. If the gaps grow in a pattern (4, 8, 16 or 2, 4, 6), the rule is one level deeper. If the gaps grow fast, try the ratio between terms or compare with squares and cubes.

In a wrong term question, one term does not fit while the others do. Find the rule from the terms that agree with each other, then see which term breaks it. The wrong term is usually in the middle, because the first and last terms often help you fix the rule.

These questions reward pattern recognition. Knowing squares up to 25 and cubes up to 10 by heart saves a lot of time.

Key formulas to remember

Common difference series: Next term = previous term + d; If successive differences are equal, the series is an arithmetic pattern. The nth term is a + (n − 1)d.
Common ratio series: Next term = previous term × r; If successive ratios are equal, the series is a geometric pattern. The nth term is a × r^(n − 1).
Second-difference rule: Differences of differences are constant → nth term has the form An² + Bn + C; Use this when the first differences form an arithmetic pattern such as 2, 4, 6, 8.
Squares to remember: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225; Many series are n² ± k. Check closeness to a square first.
Cubes to remember: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000; Many series are n³ ± k or n³ ± n.
Mixed operation rule: Next term = previous term × a + b (or × a − b); Examples: ×2 + 1 gives 3, 7, 15, 31. Test it when the differences double and the ratio is close to a whole number.
Product-type rule: n(n + 1): 2, 6, 12, 20, 30, 42, 56; Each term is the product of two consecutive numbers. Differences go 4, 6, 8, 10 and so on.
Alternating series: Odd-position terms follow one rule; even-position terms follow another; Split the series into two interleaved series when the single rule fails.

How to solve Number Series: Missing and Wrong Terms questions

Use this order for any missing-term or wrong-term question. Do not guess the rule from one pair of terms. Confirm it on every term.

1Read the question. Is a term missing (marked ?) or do you have to find the wrong term?
2Write the gaps between consecutive terms. Check if they are equal or follow a pattern.
3If the gaps are not clear, check the ratios. Also compare each term with nearby squares and cubes.
4If still no pattern, try a mixed rule (×a + b) or split the series into odd and even positions.
5Write your guessed rule and test it on every given term, not just the first two.
6For a missing term, apply the rule once more to get the value.
7For a wrong term, find the one term that fails, and note what it should have been. Pick the option that is the wrong term as given.
8Match your value with the options. If it is not there, recheck the rule.

Quickest way: Differences first, then squares and cubes

When to use it: Use this in the MCQ paper, where you have about a minute per question and wrong answers cost 0.25 marks.

Compute first differences mentally in about 10 seconds. Equal gaps or a visible pattern means you are done.
If the gaps double or triple, think ×2 + 1, ×3 − 1 and so on. Test with the first two terms.
If terms rise quickly and look like 10, 17, 26 or 7, 26, 63, think n² + 1 or n³ − 1.
For wrong-term questions, find the rule from the terms near the end and the start, then check the middle.
Use options: if only one option fits the pattern for the next gap, you can pick it without full calculation.
If no rule appears in about 60 seconds, skip. A blind guess carries a risk of 0.25 negative marks, so come back only if time remains.

Common mistakes in Number Series: Missing and Wrong Terms

Fixing the rule from only the first two terms
Several rules can fit the first few terms. Take 2, 4, 8. The rule ×2 fits it, and the next term is 16. A second rule, where the differences increase by 2, also fits. The differences of 2, 4, 8 are 2 and 4, so the next difference is 6 under this rule, and the next term is 8 + 6 = 14. Two different rules fit the same three terms but give different next terms (16 and 14).
Fix: Test your rule on at least three or four pairs before using it.
Treating a correct term as wrong
Students pick the term that looks odd instead of the one that breaks the confirmed rule.
Fix: Find the rule from the terms that agree, then check which term fails. The wrong term is the one that fails, not the one that looks unusual.
Missing the second level of differences
The first differences are not equal, so students give up on the difference method.
Fix: Take differences of the differences. If those are constant or follow a pattern, you have the rule.
Ignoring alternating patterns
Students try a single rule for the whole series even when odd and even positions behave differently.
Fix: If one rule keeps failing, write the series as two lists (positions 1, 3, 5 and 2, 4, 6) and check each.
Arithmetic slips with squares and cubes
Pressure and half-remembered tables lead to errors like 13² = 159.
Fix: Learn squares to 25 and cubes to 10. Re-check the value of the missing term against the rule once before marking.
Giving the corrected value instead of the wrong term
Students find that the wrong term should be 48 and mark 48, but the options list the given terms.
Fix: Read the question again. The wrong term is the number printed in the series. The correct value is only a check.

Worked examples

Example 1

Find the missing term: 3, 7, 15, 31, 63, ?

(A) 117 (B) 125 (C) 127 (D) 129

Show the solution

Differences: 7 − 3 = 4, 15 − 7 = 8, 31 − 15 = 16, 63 − 31 = 32.
The differences double each time: 4, 8, 16, 32. The next difference is 64.
Next term = 63 + 64 = 127.
Check with the mixed rule ×2 + 1: 3 × 2 + 1 = 7, 7 × 2 + 1 = 15, 15 × 2 + 1 = 31, 31 × 2 + 1 = 63, and 63 × 2 + 1 = 127.

Answer: (C) 127

Example 2

Find the wrong term: 3, 6, 12, 24, 50, 96, 192

(A) 6 (B) 24 (C) 50 (D) 96

Show the solution

Ratios: 6 ÷ 3 = 2, 12 ÷ 6 = 2, 24 ÷ 12 = 2. The rule looks like ×2.
Ratios 96→192 is 2 and 6→12→24 is 2, so the rule is ×2. After 24 the term should be 48, not 50; 48 × 2 = 96 and 96 × 2 = 192 confirm the rest.
Every other term fits ×2, so 50 is the wrong term.

Answer: (C) 50

Example 3

Find the missing term: 0, 7, 26, 63, 124, ?

(A) 205 (B) 210 (C) 215 (D) 225

Show the solution

The terms are close to cubes: 1, 8, 27, 64, 125 are 1³ to 5³.
Each term is one less than a cube: 1³ − 1 = 0, 2³ − 1 = 7, 3³ − 1 = 26, 4³ − 1 = 63, 5³ − 1 = 124.
The next term is 6³ − 1 = 216 − 1 = 215.

Answer: (C) 215

Exam tips

Learn squares to 25 and cubes to 10 cold. Many questions are n² ± k or n³ ± k, and recognising them takes only seconds.
Always check the rule on the whole series. Wrong-term questions are built so that a partial rule gives a tempting but wrong answer.
Use the options. In a missing-term question, compute the likely gap and see which option matches it. This saves time.
Do not spend more than about a minute on one question. Skip it and move on, as a wrong guess costs 0.25 marks.
Practise mixed rules (×2 + 1, ×3 − 2) and alternating series. They are the usual traps in the harder questions.

Practice questions from Number Series, Coding-Decoding and Odd Man Out

Number Series: Missing and Wrong Terms: frequently asked questions

How do I find the wrong number in a series quickly?

Find the rule from the terms that clearly agree with each other, often the first few and last few. Then check each middle term against it. The term that fails is the wrong one. Always confirm that all other terms fit the rule.

What patterns come up most in number series questions?

The common ones are constant difference, constant ratio, growing differences, squares, cubes, n(n + 1) type products, mixed rules like ×2 + 1, and alternating series. Check them roughly in that order.

Is there negative marking for number series in CA Foundation?

Yes. Paper 3 (Quantitative Aptitude) is an objective paper with 0.25 marks deducted for each wrong answer. If you cannot find the rule after a short try, skip the question.

What if the differences are not equal?

Take the differences of the differences. If they are constant, the series is quadratic. Also try ratios, squares, cubes and mixed rules. If the terms alternate in behaviour, split them into odd and even positions.