Sequence Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Sequence.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

An ordered list of numbers generated by a rule, where each number has a specific position (first, second, third, ...).

A pattern of numbers: first term, second term, third term, and so on.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A sequence is an ordered list of numbers defined by a rule; each term has a specific position (index).

Common stuck point: A sequence can converge (approach a limit) or diverge (grow without bound).

Sense of Study hint: Write out the first 5 or 6 terms explicitly to spot the pattern before trying to find a formula.

Worked Examples

Example 1

easy
Write the first five terms of the sequence a_n = \frac{n}{n+1} for n = 1, 2, 3, \ldots

Solution

  1. 1
    a_1 = \frac{1}{2}.
  2. 2
    a_2 = \frac{2}{3}.
  3. 3
    a_3 = \frac{3}{4}.
  4. 4
    a_4 = \frac{4}{5}.
  5. 5
    a_5 = \frac{5}{6}.

Answer

\frac{1}{2},\; \frac{2}{3},\; \frac{3}{4},\; \frac{4}{5},\; \frac{5}{6}
Each term is found by substituting n into the formula. The terms increase and approach 1, illustrating that this sequence converges to 1 as n \to \infty.

Example 2

medium
Determine whether the sequence a_n = \frac{3n^2 + 1}{n^2 + 2} converges or diverges. If it converges, find the limit.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Write the first four terms of b_n = (-1)^n \cdot \frac{1}{n}.

Example 2

medium
Determine whether a_n = \frac{2^n}{n!} converges or diverges.
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