Sequence and Series Examples: Key Types, Formulas, and Applications

When you list numbers that follow a rule, you’re working with sequences. When you add those numbers together, that’s a series. A sequence just means you’ve got a list of terms with a pattern, while a series is the sum of those terms. If you get the difference, you can figure out the nth term or the total for an arithmetic or geometric progression.

A workspace with an open notebook showing handwritten mathematical sequences and series formulas, a calculator, pencil, and ruler on a desk with a laptop in the background.

This post walks you through examples of common sequences and their matching series. I’ll point out key formulas and work through problems you might see on homework or in puzzles.

You’ll get simple examples, step-by-step calculations, and some quick tips for spotting patterns or summing terms. I hope you’ll feel a bit more confident by the end.

Essential Sequence and Series Examples

A workspace with a notebook showing mathematical formulas, a calculator, pens, and a laptop displaying graphs on a wooden desk.

Let’s look at some concrete examples. I’ll show how to find terms, write sums, and use formulas for different progressions.

Each example gives the first few terms, the rule, and how to get a specific term or sum.

Arithmetic Sequence and Series Examples

In an arithmetic sequence, you add the same number each time. If the first term a = 3 and the common difference d = 5, the first five terms are 3, 8, 13, 18, 23.

To find the nth term, use an = a + (n − 1)d. For the 8th term: a8 = 3 + 7·5 = 38.

An arithmetic series means you add the terms of an arithmetic sequence. The sum of the first n terms is Sn = (n/2)(2a + (n − 1)d).

For example, the sum of the first 6 terms of the sequence above: S6 = (6/2)(2·3 + 5·5) = 3(6 + 25) = 93.

Quick reference:

  • First term: a = 3
  • Common difference: d = 5
  • nth term: an = a + (n−1)d
  • Sum formula: Sn = (n/2)(2a + (n−1)d)

Geometric Sequence and Series Examples

A geometric sequence multiplies by a fixed ratio r. If a = 2 and r = 3, the terms are 2, 6, 18, 54, 162.

Use an = a·r^(n−1). For the 3rd term: a3 = 2·3^2 = 18.

For the 8th term: a8 = 2·3^7 = 2·2187 = 4374.

A finite geometric series sums a, ar, ar^2, … ar^(n−1). When r ≠ 1, use Sn = a(1 − r^n)/(1 − r).

Example: sum of the first 4 terms here: S4 = 2(1 − 3^4)/(1 − 3) = 2(1 − 81)/(−2) = 80.

For an infinite geometric series with |r| < 1, use S∞ = a/(1 − r). This only works when the series converges.

Harmonic Sequence and Series Examples

A harmonic sequence takes reciprocals of an arithmetic sequence. If the underlying arithmetic sequence is 1, 2, 3, 4, … then the harmonic sequence is 1, 1/2, 1/3, 1/4, …

Terms: a1 = 1, a2 = 1/2, a3 = 1/3. The nth term is 1/n.

A harmonic series adds those reciprocals: 1 + 1/2 + 1/3 + 1/4 + … This is an infinite series that diverges, so its sum grows without bound.

You can check partial sums to see divergence: S1 = 1, S2 = 1.5, S4 ≈ 2.083, S8 ≈ 2.717. The sums just keep getting bigger.

Some special cases:

  • The Fibonacci sequence isn’t arithmetic, geometric, or harmonic. It’s defined recursively (1, 1, 2, 3, 5, …).
  • A harmonic progression uses terms 1/(a + (n−1)d) when the base is an arithmetic progression.

Key Sequence and Series Formulas with Example Problems

Here are some direct formulas to get any term, compute finite sums, and handle special cases like infinite geometric sums and harmonic-like progressions.

The examples use a first term a, a common difference d, and a common ratio r so you can plug in your own numbers.

Finding the Nth Term and Specific Terms

Use the arithmetic nth-term formula when each term increases by a fixed amount. For an arithmetic sequence with first term a and common difference d, the nth term is:

  • an = a + (n − 1)d

Example: If a = 3 and d = 4, then a5 = 3 + 4·4 = 19.

Use the geometric nth-term formula when each term multiplies by the same ratio. For a geometric sequence with first term a and common ratio r, the nth term is:

  • an = a · r^(n−1)

Example: If a = 5 and r = 2, then a4 = 5 · 2^3 = 40.

You can use summation notation (Σ) to write either sequence compactly. For example, Σ_{k=1}^{n} a·r^{k−1} lists the first n terms of a geometric sequence.

Summation of Finite and Infinite Series

For an arithmetic series, sum the first n terms with:

  • Sn = n/2 · (2a + (n − 1)d) or Sn = n/2 · (a + an)

Example: a = 1, d = 3, n = 4 gives S4 = 4/2 · (2·1 + 3·3) = 2 · 11 = 22.

For a finite geometric series, use:

  • Sn = a · (1 − r^n) / (1 − r) if r ≠ 1

Example: a = 2, r = 1/3, n = 3 gives S3 = 2·(1−(1/3)^3)/(1−1/3) = 2·(1−1/27)/(2/3) = 2·(26/27)·(3/2) = 26/9.

For an infinite geometric series, the sum exists only when |r| < 1:

  • S = a / (1 − r)

Example: a = 4, r = 1/2 gives S = 4 / (1 − 1/2) = 8.

Special Series and Progression Applications

Harmonic-type sequences flip an arithmetic sequence by taking reciprocals, so the terms show up as 1/(a + (n − 1)d).

You won’t find a neat, closed formula for their sums like you do with arithmetic or geometric series. Sometimes you can approximate these sums or, if you’re lucky, use logarithms for certain cases.

People often use arithmetic mean–geometric mean concepts when working with inequalities or comparing different sequences.

In real-world problems, you’ll usually end up mixing these types. Take compound interest—it relies on geometric sums. Meanwhile, if you’re figuring out totals for evenly spaced payments, you’ll use arithmetic sums.

Here’s a quick reference:

  • Arithmetic nth term: an = a + (n−1)d
  • Arithmetic sum: Sn = n/2(2a + (n−1)d)
  • Geometric nth term: an = a·r^(n−1)
  • Geometric sum (finite): Sn = a(1−r^n)/(1−r)
  • Geometric sum (infinite): S = a/(1−r) for |r|<1

Just match the formula to your sequence, plug in your values for a, d, or r, and you’re set to solve the problem.

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