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Quantitative Aptitude · Sequence and Series

Concept of Sequence and Series for CA Foundation

Updated 1 October 2026 · Fact-checked

A sequence is a list of numbers arranged in a definite order, each following a rule. A series is the sum of the terms of a sequence. To solve questions, find the rule for the nth term, put in the needed value of n, and add terms if a series is asked.

Understand Concept of Sequence and Series

A sequence is a set of numbers written in a fixed order, where each number is called a term. The position of a term is its place in the list. The first term is t₁ (or a₁), the second is t₂, and so on. Example: 2, 4, 6, 8, ... Here t₁ = 2, t₂ = 4.

Most sequences follow a rule. The rule that gives the term at position n is called the general term or nth term, written tₙ. For 2, 4, 6, 8, ..., tₙ = 2n. Put n = 5 and you get t₅ = 10. Once you have tₙ, you can get any term without listing the ones before it.

A series is what you get when you add the terms of a sequence. For the sequence 2, 4, 6, 8, the series is 2 + 4 + 6 + 8. Its value is 20. The sum of the first n terms is written Sₙ. So Sₙ = t₁ + t₂ + ... + tₙ.

A sequence with a last term is finite, like 3, 6, 9, 12. A sequence that goes on without end is infinite, like 1, 2, 3, ... The same words apply to series: a finite series has a last term, an infinite series does not.

The key link between the two is this: the nth term equals the sum of n terms minus the sum of (n − 1) terms. That is tₙ = Sₙ − Sₙ₋₁ for n ≥ 2, and t₁ = S₁. Exams use this link often.

Key formulas to remember

General term
tₙ = rule in n (e.g. tₙ = 3n + 2)
Put n = 1, 2, 3, ... to generate terms. n is a positive integer.
Series as a sum
Sₙ = t₁ + t₂ + ... + tₙ
Sₙ is the sum of the first n terms.
Term from sum
tₙ = Sₙ − Sₙ₋₁ (for n ≥ 2); t₁ = S₁
Use when the sum formula is given and you need a particular term.
Sum from terms
Sₙ = Sₙ₋₁ + tₙ
Use when you know the sum up to n − 1 and the nth term.

How to solve Concept of Sequence and Series questions

Use this method for any basic question on sequences and series.

  1. 1Read the question and decide: is it asking for a term (sequence) or a sum (series)?
  2. 2Write down the given rule: tₙ or Sₙ. If only a few terms are given, look for the pattern first.
  3. 3For a term, substitute the required value of n into tₙ.
  4. 4For a term when Sₙ is given, compute Sₙ − Sₙ₋₁ with n replaced properly. Use S₁ for the first term.
  5. 5For a sum of a few terms, find each term and add them.
  6. 6Check whether the sequence is finite or infinite if the question asks for it: look for a last term.
  7. 7Test your answer against the options by substituting back.

Quickest way: Substitute and eliminate

When to use it: Use in MCQs where tₙ or Sₙ is given and options are numbers or formulas.

  1. If options are values, put the small n directly into the given formula. Do not simplify first.
  2. If options are formulas for tₙ, test n = 1 and n = 2 against the given first terms. Cross out options that fail.
  3. For tₙ from Sₙ, find S₁, S₂ and use t₂ = S₂ − S₁. Match with the options.
  4. Skip a question if finding the pattern takes more than about a minute. A wrong answer costs 0.25 marks.

Common mistakes in Concept of Sequence and Series

  • Treating sequence and series as the same thing.

    Both use the same numbers, and everyday language mixes them.

    Fix: Sequence is a list separated by commas. Series is the terms joined by plus signs, or their sum.

  • Finding tₙ from Sₙ by subtracting S(n − 1) wrongly, such as writing Sₙ − 1.

    Students read Sₙ₋₁ as Sₙ minus 1.

    Fix: Replace every n in the formula for Sₙ with (n − 1), then subtract.

  • Using tₙ = Sₙ − Sₙ₋₁ for the first term.

    The formula looks general.

    Fix: S₀ is not defined here. Use t₁ = S₁.

  • Starting n from 0 instead of 1.

    Habit from other subjects.

    Fix: The first term is at n = 1. Check by writing t₁ first.

  • Calling a sequence infinite because it has dots, e.g. 2, 4, 6, ..., 20.

    Dots are assumed to mean no end.

    Fix: Look for a last term. If one is written after the dots, the sequence is finite.

Worked examples

Example 1

The nth term of a sequence is tₙ = 3n + 2. What is the 10th term? (a) 30 (b) 32 (c) 35 (d) 23

Show the solution
  1. Put n = 10 in tₙ = 3n + 2.
  2. t₁₀ = 3 × 10 + 2 = 30 + 2 = 32.

Answer: (b) 32

Example 2

The sum of the first n terms of a series is Sₙ = n² + 2n. What is the 5th term? (a) 35 (b) 11 (c) 13 (d) 9

Show the solution
  1. t₅ = S₅ − S₄.
  2. S₅ = 25 + 10 = 35.
  3. S₄ = 16 + 8 = 24.
  4. t₅ = 35 − 24 = 11.
  5. Check: general tₙ = (n² + 2n) − ((n − 1)² + 2(n − 1)) = 2n + 1. For n = 5 this gives 11.

Answer: (b) 11

Example 3

Which one of these is a finite sequence? (a) 1, 3, 5, 7, ... (b) 2, 4, 8, 16, ... (c) 5, 10, 15, ..., 100 (d) 1, 1/2, 1/4, ...

Show the solution
  1. A finite sequence has a last term.
  2. Options (a), (b) and (d) end with dots and no last term, so they continue forever.
  3. Option (c) ends with 100, so it stops there.

Answer: (c) 5, 10, 15, ..., 100

Exam tips

  • Expect direct questions: find a particular term from tₙ, or find tₙ from Sₙ. These are quick marks.
  • Always check the first term when you are given Sₙ. Compute S₁ and compare.
  • Read whether the question says 'term' or 'sum of terms'. Options often include both values.
  • Questions on this topic are short. Do them first and save time for longer calculation questions.
  • Sequence and series basics lead into AP and GP. Learn the notation tₙ and Sₙ well now.

Practice questions from Sequence and Series

Concept of Sequence and Series: frequently asked questions

What is the difference between sequence and series?

A sequence is an ordered list of terms, like 2, 4, 6, 8. A series is the sum of those terms, like 2 + 4 + 6 + 8 = 20. The sequence lists the terms and the series adds them.

What is the nth term of a sequence?

It is a formula that gives the term at position n. For example, if tₙ = 2n + 1, then t₃ = 7. It lets you find any term directly.

How do I find the nth term from the sum of n terms?

Use tₙ = Sₙ − Sₙ₋₁ for n ≥ 2. The first term is S₁. Replace n by (n − 1) in the sum formula before subtracting.

How do I tell a finite sequence from an infinite one?

A finite sequence has a last term, so you can count its terms. An infinite sequence never ends. A list written with dots at the end and no final term is infinite.