Sequence Formula

The Formula

\{a_n\}_{n=1}^{\infty} = a_1, a_2, a_3, \ldots \quad \text{Converges if } \lim_{n \to \infty} a_n = 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

a_n = 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 : \mathbb{N} \to \mathbb{R}, written (a_n)_{n=1}^{\infty}. The sequence converges to L if \forall \epsilon > 0,\; \exists N \in \mathbb{N} : n > N \implies |a_n - L| < \epsilon.

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.

Common Mistakes

  • Confusing a sequence with a series: a sequence is a list of numbers (a_1, a_2, a_3, \ldots), while a series is the sum of those numbers (a_1 + a_2 + a_3 + \ldots).
  • Writing the general term formula with wrong indexing: if the sequence starts at n = 1 with value 3, the formula a_n = 2n + 1 gives a_1 = 3 but a_n = 2n gives a_1 = 2 — always verify with the first few terms.
  • Assuming a sequence that increases must diverge: the sequence a_n = 1 - \frac{1}{n} is increasing but converges to 1.

Why This Formula Matters

Sequences are the foundation for series, limits, convergence, and patterns in higher mathematics.

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?

a_n = nth term

Why is the Sequence formula important in Math?

Sequences are the foundation for series, limits, convergence, and patterns in higher mathematics.

What do students get wrong about Sequence?

A sequence can converge (approach a limit) or diverge (grow without bound).

Want the Full Guide?

This formula is covered in depth in our complete guide:

Growing Patterns, Arithmetic and Geometric Sequences →
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