Sequences & Series: Arithmetic, Geometric & Convergence

What Are Sequences and Series?

A sequence is an ordered list of numbers following a rule. A series is the sum of the terms of a sequence. These concepts appear everywhere in mathematics — from simple patterns in number theory to compound interest in finance and convergence tests in calculus.

Learning Goals: By the end of this guide, you should be able to:

1. Identify and generate arithmetic and geometric sequences from their rules.
2. Find the $n$th term and the sum of the first $n$ terms for both types.
3. Determine whether an infinite geometric series converges, and if so, find its sum.
4. Apply sequences and series to real-world problems.

Arithmetic Sequences

An arithmetic sequence has a constant difference $d$ between consecutive terms.

Key Formulas

$n$th term:

Sum of first $n$ terms:

Example

The sequence $3, 7, 11, 15, \ldots$ has $a_1 = 3$ and $d = 4$.

• $a_{10} = 3 + 9 \times 4 = 39$
• $S_{10} = \frac{10}{2}(3 + 39) = 210$

Geometric Sequences

A geometric sequence has a constant ratio $r$ between consecutive terms.

Key Formulas

$n$th term:

Sum of first $n$ terms (when $r \neq 1$):

Sum to infinity (only when $|r| < 1$):

Example

The sequence $2, 6, 18, 54, \ldots$ has $a_1 = 2$ and $r = 3$.

• $a_6 = 2 \times 3^5 = 486$
• $S_6 = 2 \times \frac{1 - 3^6}{1 - 3} = 2 \times \frac{-728}{-2} = 728$

Convergence of Infinite Series

An infinite geometric series $a_1 + a_1 r + a_1 r^2 + \cdots$ converges (has a finite sum) if and only if $|r| < 1$.

Example: Sum to Infinity

Find the sum of $8 + 4 + 2 + 1 + \cdots$

Here $a_1 = 8$, $r = \frac{1}{2}$. Since $|r| < 1$:

Worked Example: Real-World Application

Question: A ball is dropped from a height of 10 m. Each bounce reaches 60% of the previous height. Find the total distance travelled before the ball comes to rest.

Solution: The ball falls 10 m, then bounces up $10 \times 0.6 = 6$ m, falls 6 m, bounces up $6 \times 0.6 = 3.6$ m, and so on.

Total distance $= 10 + 2(6 + 3.6 + 2.16 + \cdots) = 10 + 2 \times \frac{6}{1 - 0.6} = 10 + 2 \times 15 = 40$ m.

Common Mistakes

1. Off-by-one errors in the $n$th term formula — Remember: $a_n = a_1 + (n-1)d$, not $a_1 + nd$. The first term is $a_1$, not $a_0$.
2. Using $S_\infty$ when $|r| \geq 1$ — The sum to infinity formula is only valid for convergent series. Always check $|r| < 1$ first.
3. Confusing $a_n$ with $S_n$ — The $n$th term is a single number; the sum $S_n$ is the total of the first $n$ terms.

Exam Tips (A-Level / AP / IB)

• If given two terms (e.g., $a_3 = 12$ and $a_7 = 48$), set up simultaneous equations to find $a_1$ and $d$ (or $r$).
• For "prove that $S_n = ...$" questions, use mathematical induction or the technique of writing $S_n - S_{n-1} = a_n$.
• In MCQ format, quickly test whether a series converges by checking if $|r| < 1$.

Frequently Asked Questions

What is the difference between a sequence and a series?

A sequence is a list: $2, 4, 6, 8, \ldots$ A series is the sum: $2 + 4 + 6 + 8 + \cdots$ The sequence gives individual terms; the series gives their running total.

Can an arithmetic series ever converge?

No — unless $d = 0$ and $a_1 = 0$, the terms of an arithmetic sequence do not approach zero, so the series always diverges.

Related Topics

• Derivatives from First Principles — Limits are the foundation of both convergence and differentiation.
• Probability Distributions — The geometric distribution is closely related to geometric sequences.
• Linear Regression — Summing squared residuals uses series.