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UGC NET PAPER 1 · UNIT 05 · MODULE 2: REASONING, SERIES, CODES AND RELATIONS · TOPIC 2

Number Series

A four-step method, the pattern catalogue and the sum formulas that series questions rely on.

  • 6 sub-topics
  • 6 min read
  • Topic 2 of 17

What this topic covers

6 sub-topics, in the order to study them. Tap one to jump to it.

  1. §OverviewA number series is a list of numbers that follows a hidden rule.
  2. 2.1Four-step method (use it every time)Covers Look at the growth, Write the differences, If the differences are not constant, If differences fail.
    • Look at the growth
    • Write the differences
    • If differences fail
  3. 2.2Pattern catalogue with examples# Pattern Example Next term Rule 1 Arithmetic (constant difference) 5, 9, 13, 17 21 +4 2 Geometric (constant ratio) 3, 6, 12, 24 48 x2 3 Squares 4, 9, 16, 25 36 n squared 4…
  4. 2.3Standard formulas for series and sums (memory questions)Item Formula nth term of an AP (first term a, common difference d) a + (n - 1)d Sum of n terms of an AP n/2 x [2a + (n - 1)d], or n/2 x (first + last) nth term of a GP…
  5. 2.4Shortcuts and tricksCovers Near a square or cube?, Fast growth test, Slow, irregular growth, Zig-zag numbers, Check the last digit and more.
    • Near a square or cube?
    • Fast growth test
    • Slow, irregular growth
    • Zig-zag numbers
    • Check the last digit
  6. 2.5Worked examplesExample 1. 4, 7, 12, 19, 28, ?
    • Example 1
    • Example 2
    • Example 3
    • Example 4
    • Example 5
  7. 2.6TrapsUsing only the first two terms to guess a rule, which often fits several different patterns. Forgetting that the series may be in descending order. Missing that a series is…

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§Overview#

A number series is a list of numbers that follows a hidden rule. You are asked to find the missing term, the wrong term, or the odd one out. Most questions use one of about a dozen patterns, so recognising the pattern quickly is the whole skill.

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2.1Four-step method (use it every time)#

In brief

Covers Look at the growth, Write the differences, If the differences are not constant, If differences fail.

  • Look at the growth
  • Write the differences
  • If differences fail

  1. Look at the growth. If the numbers rise slowly, think addition or small differences. If they rise very fast, think multiplication, squares, cubes or powers.

  2. Write the differences between neighbouring terms. If the differences are constant, it is an arithmetic series.

  3. If the differences are not constant, write the differences of the differences, or check whether the differences are squares, primes, or double each time.

  4. If differences fail, test ratios, then squares and cubes (with a small plus or minus), then alternating or two-series patterns.

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2.2Pattern catalogue with examples#

In brief

# Pattern Example Next term Rule 1 Arithmetic (constant difference) 5, 9, 13, 17 21 +4 2 Geometric (constant ratio) 3, 6, 12, 24 48 x2 3 Squares 4, 9, 16, 25 36 n squared 4…

# Pattern Example Next term Rule
1 Arithmetic (constant difference) 5, 9, 13, 17 21 +4
2 Geometric (constant ratio) 3, 6, 12, 24 48 x2
3 Squares 4, 9, 16, 25 36 n squared
4 Cubes 8, 27, 64, 125 216 n cubed
5 n squared plus a constant 2, 5, 10, 17 26 n squared + 1
6 n squared minus a constant 3, 8, 15, 24 35 n squared - 1
7 n(n+1) (product of consecutive numbers) 2, 6, 12, 20 30 n(n+1)
8 n cubed minus n 0, 6, 24, 60 120 n cubed - n
9 Triangular numbers 1, 3, 6, 10, 15 21 n(n+1)/2
10 Difference rises by a constant (second order) 2, 5, 11, 20, 32 47 differences 3, 6, 9, 12, 15
11 Differences are squares 1, 2, 6, 15, 31 56 differences 1, 4, 9, 16, 25
12 Differences double 3, 4, 6, 10, 18 34 differences 1, 2, 4, 8, 16
13 Multiply and add 3, 7, 15, 31 63 x2 + 1
14 Multiply and add (x3) 1, 4, 13, 40 121 x3 + 1
15 Multiplier increases 1, 2, 6, 24, 120 720 x1, x2, x3, x4, x5, x6
16 Multiply and add increasing amounts 3, 4, 10, 33, 136 685 x1+1, x2+2, x3+3, x4+4, x5+5
17 Primes 11, 13, 17, 19 23 consecutive primes
18 Primes plus or minus a constant 4, 6, 8, 12, 14 18 prime + 1
19 Fibonacci type (sum of previous two) 1, 2, 3, 5, 8, 13 21 previous two added
20 Alternating squares and cubes 1, 4, 27, 16, 125, 36 343 1 squared, 2 squared, 3 cubed, 4 squared, 5 cubed, 6 squared, 7 cubed
21 Two interleaved series 2, 5, 4, 8, 6, 11, 8 14 odd places +2, even places +3
22 Alternating operations 5, 10, 8, 16, 14, 28 26 x2 then -2
23 Decreasing n squared minus 1 120, 99, 80, 63, 48 35 11 squared - 1, 10 squared - 1, and so on

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2.3Standard formulas for series and sums (memory questions)#

In brief

Item Formula nth term of an AP (first term a, common difference d) a + (n - 1)d Sum of n terms of an AP n/2 x [2a + (n - 1)d], or n/2 x (first + last) nth term of a GP…

Item Formula
nth term of an AP (first term a, common difference d) a + (n - 1)d
Sum of n terms of an AP n/2 x [2a + (n - 1)d], or n/2 x (first + last)
nth term of a GP (common ratio r) a x r^(n-1)
Sum of n terms of a GP (r not equal to 1) a(r^n - 1)/(r - 1)
Sum of first n natural numbers n(n + 1)/2
Sum of squares of first n natural numbers n(n + 1)(2n + 1)/6
Sum of cubes of first n natural numbers [n(n + 1)/2] squared
Sum of first n odd numbers n squared
Sum of first n even numbers n(n + 1)
Number of terms in an AP (last - first)/d + 1

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2.4Shortcuts and tricks#

In brief

Covers Near a square or cube?, Fast growth test, Slow, irregular growth, Zig-zag numbers, Check the last digit and more.

  • Near a square or cube?
  • Fast growth test
  • Slow, irregular growth
  • Zig-zag numbers
  • Check the last digit
  • Decreasing series

  • Memorise squares up to 30 and cubes up to 15. Almost every "hidden" square or cube pattern is solved instantly when you recognise 121, 169, 196, 289, 343, 512, 729 and similar numbers.

  • Near a square or cube? If the terms sit just above or below perfect squares or cubes, test n squared +/- k and n cubed +/- k.

  • Fast growth test: if each term is roughly double or triple the previous one, test x2 +/- c or x3 +/- c before anything else.

  • Slow, irregular growth: write the differences; if differences themselves grow steadily, go to second-order differences.

  • Zig-zag numbers (up, down, up, down) usually mean two alternating operations or two interleaved series.

  • Check the last digit. If the last digits repeat in a cycle such as 2, 4, 8, 6, suspect powers or repeated doubling.

  • Test the rule on two terms before you trust it. A rule that fits only the first two terms is usually wrong.

  • Decreasing series: try reverse of the increasing patterns (minus, divide, descending squares or cubes).

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2.5Worked examples#

In brief

Example 1. 4, 7, 12, 19, 28, ?

  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Example 5

Example 1. 4, 7, 12, 19, 28, ?

  • Differences: 3, 5, 7, 9. These rise by 2, so the next difference is 11.

  • Answer: 28 + 11 = 39. Check: the terms are n squared + 3 and 6 squared + 3 = 39.

Example 2. 6, 11, 21, 36, 56, ?

  • Differences: 5, 10, 15, 20, so the next is 25.

  • Answer: 56 + 25 = 81.

Example 3. 3, 7, 15, 31, 64, 127. Find the wrong term.

  • Rule from the first terms: x2 + 1 gives 7, 15, 31, 63, 127.

  • The term 64 breaks the rule. The correct term is 63, so the wrong term is 64.

Example 4. 3, 5, 7, 12, 13, 17, 19. Find the odd one out.

  • All terms except 12 are prime numbers. The odd one out is 12.

Example 5. 2, 5, 4, 8, 6, 11, 8, ?

  • Odd places: 2, 4, 6, 8 (adding 2). Even places: 5, 8, 11, so the next is 14.

  • Answer: 14.

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2.6Traps#

In brief

Using only the first two terms to guess a rule, which often fits several different patterns. Forgetting that the series may be in descending order. Missing that a series is…

  • Using only the first two terms to guess a rule, which often fits several different patterns.

  • Forgetting that the series may be in descending order.

  • Missing that a series is two interleaved series because the numbers look random.

  • Reading "wrong number" questions as "missing number" questions. For a wrong-number question, find the rule from the early, reliable terms and locate where it breaks.

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Test yourself on Number Series

Practise with free MCQs and answer keys, then return here to fix what you got wrong.

↑ Module 2: Reasoning, Series, Codes and Relations · All 18 topics in Mathematical Reasoning · Unit overview and sample MCQs

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