Sequence Formula

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

The Formula

{an}n=1∞=a1,a2,a3,…Converges if lim⁡n→∞an=L

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

Quick Example

1, 4, 9, 16, 25... (squares). 1, 1, 2, 3, 5, 8... (Fibonacci).

Notation

an = nth term

What This Formula Means

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.

Formal View

A sequence is a function a:N→R, written (an)n=1∞. The sequence converges to L if ∀ϵ>0,  ∃N∈N:n>N  ⟹  ∣an−L∣<ϵ.

Worked Examples

Example 1

easy
Write the first five terms of the sequence an=nn+1 for n=1,2,3,…

Answer

12,  23,  34,  45,  56

First step

1
a1=12.

Full solution

  1. 2
    a2=23.
  2. 3
    a3=34.
  3. 4
    a4=45.
  4. 5
    a5=56.
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→∞.

Example 2

medium
Determine whether the sequence an=3n2+1n2+2 converges or diverges. If it converges, find the limit.

Example 3

easy
Show step-by-step output for: n=2; n=n∗3; n=n+1; OUTPUT n.

Common Mistakes

  • Treating a sequence as a sum — listing terms is a sequence; only adding them makes a series.
  • Ignoring order — a sequence is ordered, so 2,5,8 and 8,5,2 are different sequences.
  • Saying a sequence converges when its terms keep growing — convergence requires lim⁡n→∞an to be a finite value.

Why This Formula Matters

Sequences are the raw material for series, limits at infinity, and convergence — the language for anything that proceeds step by step. The single most important distinction students must hold is sequence (a list of terms) versus series (their sum); confusing the two derails every later convergence question. Recognizing it by "Am I listing terms by position (an) rather than adding them up?" — rather than by familiar numbers — is what lets a student tell it apart from series and function and set in a mixed problem set.

Frequently Asked Questions

What is the Sequence formula?

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

How do you use the Sequence formula?

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

What do the symbols mean in the Sequence formula?

an = nth term

Why is the Sequence formula important in Math?

Sequences are the raw material for series, limits at infinity, and convergence — the language for anything that proceeds step by step. The single most important distinction students must hold is sequence (a list of terms) versus series (their sum); confusing the two derails every later convergence question. Recognizing it by "Am I listing terms by position (an) rather than adding them up?" — rather than by familiar numbers — is what lets a student tell it apart from series and function and set in a mixed problem set.

What do students get wrong about Sequence?

The procedure for sequence is the easy part; the trap is treating a sequence as a sum. Asking "Am I listing terms by position (an) rather than adding them up?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Want the Full Guide?

This formula is covered in depth in our complete guide:

Growing Patterns, Arithmetic and Geometric Sequences →
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