Arithmetic and Geometric Sequence Calculator

This number sequence calculator includes arithmetic, geometric, and Fibonacci or recursive sequence calculator. In each case, the sequence calculator finds the nth term of the sequence.

Directions for use

Arithmetic sequence calculator

Use the arithmetic sequence calculator to find the nᵗʰ term of the arithmetic sequence. Enter the first number of the sequence and the common difference (usually denoted as f). Then enter the value of n to obtain the nᵗʰ number of the sequence. For example, if you need the twentieth term, enter n = 20. The calculator will return the 20ᵗʰ value and the sum of all terms up to (and including) the 20ᵗʰ term.

Geometric sequence calculator

Use the geometric sequence calculator to find the nᵗʰ term of the geometric sequence. Enter the first number of the sequence, the common ratio (usually denoted as r), and the value of n. Then press "Calculate." The calculator will return the value of the nᵗʰ term of the sequence and the sum of all numbers up to (and including) the nᵗʰ term.

Fibonacci Sequence Calculator

Use the Fibonacci sequence calculator to find the nᵗʰ term of the Fibonacci sequence. Enter the value of n, and press "Calculate." The calculator will return the nᵗʰ term of the sequence and the sum of all numbers up to (and including) the nᵗʰ value.

Definitions

Mathematical sequences

In mathematics, a number sequence is defined as a list of numbers in order. "In order" means that each number has a fixed position. A number sequence is denoted as a list of numbers separated by commas and enclosed in curly brackets. For example, {1, 3, 5, 7, 9} or {0, 1, 0, 1, 0, 1, …}.

Each sequence term is denoted as aₙ, where n – is the number of that term. For example, in the {1, 3, 5, 7, 9} sequence a₁ = 1, a₂ = 3, and so on. A number sequence usually has a rule allowing one to find any term of that sequence. The three most commonly used sequences are arithmetic, geometric, and Fibonacci.

Arithmetic sequence

The difference between any two neighboring terms is a constant in an arithmetic sequence. If we denote that constant as f, we will get aₙ₊₁ – aₙ = f, for any n. In general, any arithmetic sequence can be written as follows:

{a₁, a₁ + f, a₁ + 2f, a₁ + 3f, …}

The two important elements of any arithmetic sequence are the first term a₁, and the constant f called the common difference. Knowing these two values, we can write down the rule of the arithmetic sequence:

aₙ = a₁ + f × (n-1)

For example, let's find the 9ᵗʰ term of an arithmetic sequence with a₁ = 2 and f = 1.2. We need to find the 9ᵗʰ term. Therefore, n = 9. Using the arithmetic sequence rule, we immediately get the following:

a₉ = 2 + 1.2 × (9-1) = 2 + 1.2 × 8 = 2 + 9.6 = 11.6

Geometric sequence

In a geometric sequence, each term can be found by multiplying the previous term by a non-zero constant. That constant is usually denoted as r, called the common ratio. In a geometric sequence, aₙ₊₁ = aₙ × r. In general, any geometric sequence can be written as follows:

{a₁, a₁ × r, a₁ × r², a₁ × r³, …}

Knowing the first term and the common ratio, the rule of the geometric sequence can be written as follows:

aₙ = a₁ × rⁿ⁻¹

For example, let's find the 5th term of the geometric sequence with a1 = 6, and r = 2. We need to find the 5th term. Therefore, n = 5.

a₅ = a₁ × r⁵⁻¹ = 6 × 2⁴ = 6 × 16 = 96

Fibonacci sequence

Fibonacci sequence is the following sequence:

{0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, …}

In this sequence, each term is defined as the sum of two previous terms:

aₙ = aₙ₋₁ + aₙ₋₂

The first two terms of a Fibonacci sequence are commonly defined as 0 and 1.

Unlike other sequences, the Fibonacci sequence starts with a₀, not a₁! This means that a₀ = 0, a₁ = 1, a₂ = 1, a₃ = 2, and so on.

Golden ratio

The Fibonacci sequence has many interesting properties, the most notable being the golden ratio property. This property means that the ratio of any two consecutive numbers (starting with a₃ and a₄) from the Fibonacci sequence is close to the golden ratio, approximately estimated as 1.618034, and denoted as ϕ. The greater the terms of the sequence, the closer their ratio is to the golden ratio. For example,

a₄ / a₃ = 1.5

a₅ / a₄ = 1.67

a₆ / a₅ = 1.6

and so on

The golden ratio can also be used to find the terms of the Fibonacci sequence by using the following formula:

$$aₙ=\frac{φⁿ-(1-φ)ⁿ}{\sqrt{5}}$$

The more accurate value of the golden ratio you will use, the closer the calculated value of an will be to the corresponding integer of the Fibonacci sequence.

Real life example

Let's look at an example of using an arithmetic sequence in real life. Imagine you want to organize a holiday dinner at a restaurant. Usually, in this restaurant, people sit at small square tables so that four people fit at each table.

If you move two tables together, you can seat 6 people. 3 tables would seat 8 people, and so on. The restaurant only has 15 tables, and you are coming with a big group of 40 people. Will there be enough tables to seat everyone at one big joint table?

Solution

The situation above describes an arithmetic sequence with the common difference f = 2: a₁ = 4, a₂ = 6, a₃ = 8, … The restaurant only has 15 tables. Therefore, the last term of the sequence will be a₁₅. To solve the problem, we need to calculate the value of a₁₅ and compare it to the number of people – 40. Using the arithmetic sequence rule, we will get the following:

a₁₅ = a₁ + f × (15-1) = 4 + 2 × 14 = 4 + 28 = 32

Answer

Moving all tables together will only give you 32 seats, which is insufficient to put all guests at one table.