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Fundamentals of Business Mathematics and Statistics · Arithmetic Progression and Geometric Progression

Sequence and Series Basics for CMA Foundation

Updated 10 October 2026 · Fact-checked

A sequence is a list of numbers in a fixed order, such as 3, 6, 9, 12. A series is the sum of the terms of a sequence, such as 3 + 6 + 9 + 12. To solve questions, find the general term (nth term), then put in values of n to get any term.

Understand Sequences and Series Basics

A sequence is a list of numbers written in a definite order. Each number is a term. In 5, 10, 15, 20, the first term is 5, the second is 10, and so on. Order matters: 5, 10, 15 is not the same sequence as 10, 5, 15.

A series is what you get when you add the terms of a sequence. From the sequence 5, 10, 15, 20 you get the series 5 + 10 + 15 + 20 = 50. So a sequence is a list and a series is a sum. This is the most common one-line question in this topic.

Most sequences follow a rule. The rule that gives the term at position n is the general term, written Tₙ or aₙ. For example, if Tₙ = 3n + 2, then T₁ = 5, T₂ = 8, T₃ = 11. You never need to list all terms. You just put in the value of n you want.

A finite sequence has a last term, such as 2, 4, 6, 8, 10. An infinite sequence goes on without end, such as 2, 4, 6, 8, and so on. A finite sequence gives a finite series. An infinite sequence gives an infinite series, which may or may not have a fixed total.

This topic is the base for Arithmetic Progression and Geometric Progression. In those, the general term follows a fixed pattern: a constant difference or a constant ratio.

Key formulas to remember

General term of a sequence
Tₙ = f(n)
A rule in n. Put in n = 1, 2, 3 to get the first, second, third term.
Term from a sum (series)
Tₙ = Sₙ − Sₙ₋₁ (for n ≥ 2), and T₁ = S₁
Sₙ is the sum of the first n terms. Use T₁ = S₁ separately for the first term.
Sum of the first n terms
Sₙ = T₁ + T₂ + … + Tₙ
This is the series made from the first n terms.
Sum of first n natural numbers
1 + 2 + 3 + … + n = n(n + 1) ÷ 2
Useful for checking series quickly.

How to solve Sequences and Series Basics questions

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

  1. 1Read whether the question asks for a sequence (terms) or a series (sum).
  2. 2Write down the general term Tₙ if it is given. If only terms are given, look for the pattern.
  3. 3To find a particular term, replace n by that number and calculate carefully.
  4. 4To find the sum of a few terms, calculate each term and add them.
  5. 5If a sum Sₙ is given and a term is asked, use Tₙ = Sₙ − Sₙ₋₁.
  6. 6Check whether the sequence is finite or infinite, and count how many terms there are.
  7. 7Test your answer by checking it against the first one or two given terms.

Quickest way: Substitute and check options

When to use it: Use when the MCQ gives a general term or a few terms and four numeric options.

  1. If Tₙ is given, put in the n asked. Do not expand anything extra.
  2. If terms are given and you must pick the rule, put n = 1 and n = 2 into each option and discard those that fail.
  3. For a term from Sₙ, find Sₙ and Sₙ₋₁ with small numbers and subtract.
  4. Remember: sequence means list, series means sum. Eliminate options that confuse the two.

Common mistakes in Sequences and Series Basics

  • Treating sequence and series as the same thing.

    The words sound alike and are often used loosely in daily talk.

    Fix: Sequence = list of terms. Series = sum of those terms. Ask yourself if the answer should be one term or a total.

  • Using Tₙ = Sₙ − Sₙ₋₁ for n = 1.

    S₀ is not defined in the question, so the formula breaks down.

    Fix: Always take T₁ = S₁. Use the subtraction only for n ≥ 2.

  • Putting the wrong value of n, such as n = 5 for the 4th term.

    Students start counting from 0 or rush the substitution.

    Fix: The first term is n = 1. Write the n value down before calculating.

  • Calling a sequence infinite just because it has many terms.

    A long list looks endless.

    Fix: A sequence is finite if it has a last term. Infinite means no last term.

  • Sign errors when n is put into expressions such as 2n² − 3n.

    Squares and negatives are handled in a hurry.

    Fix: Put n in brackets first, for example 2(4)² − 3(4) = 32 − 12 = 20.

Worked examples

Example 1

The general term of a sequence is Tₙ = 2n² − 3. Find the sum of its first three terms.

Show the solution
  1. T₁ = 2(1)² − 3 = 2 − 3 = −1.
  2. T₂ = 2(2)² − 3 = 8 − 3 = 5.
  3. T₃ = 2(3)² − 3 = 18 − 3 = 15.
  4. S₃ = −1 + 5 + 15 = 19.

Answer: 19

Example 2

The sum of the first n terms of a series is Sₙ = n² + 2n. Find the 5th term.

Show the solution
  1. S₅ = 5² + 2(5) = 25 + 10 = 35.
  2. S₄ = 4² + 2(4) = 16 + 8 = 24.
  3. T₅ = S₅ − S₄ = 35 − 24 = 11.
  4. Check with the general term: Tₙ = (n² + 2n) − ((n−1)² + 2(n−1)) = 2n + 1, so T₅ = 11.

Answer: 11

Exam tips

  • Expect short direct questions: find a term, find a sum of a few terms, or state the difference between sequence and series.
  • Substituting n is quick. Do it before trying to find any pattern.
  • When Sₙ is given, always work out two sums with small numbers and subtract.
  • Read whether the question wants a term or a sum. Options often include both values.
  • There is no negative marking, so attempt every question even if you have to guess after eliminating options.

Practice questions from Arithmetic Progression and Geometric Progression

Sequences and Series Basics in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Sequences and Series Basics: frequently asked questions

What is the difference between a sequence and a series?

A sequence is an ordered list of numbers, like 2, 4, 6, 8. A series is the sum of the terms, like 2 + 4 + 6 + 8 = 20. The sequence gives you terms and the series gives you a total.

What is the nth term of a sequence?

It is a rule, written Tₙ, that gives the term at position n. Put n = 1 for the first term, n = 2 for the second, and so on. For example, if Tₙ = 4n − 1, then T₃ = 11.

What is the difference between a finite and an infinite sequence?

A finite sequence has a last term and a fixed number of terms. An infinite sequence has no last term and goes on endlessly.

How do I find a term if only the sum of n terms is given?

Find Sₙ and Sₙ₋₁ and subtract: Tₙ = Sₙ − Sₙ₋₁ for n ≥ 2. For the first term, use T₁ = S₁.