Find The 20th Term of The Following Sequence Calculator

Finding the nth term of a sequence is a fundamental math skill used in algebra, physics, and computer science. This calculator helps you determine the 20th term of any arithmetic or geometric sequence quickly and accurately.

Introduction

Sequences are ordered lists of numbers that follow a specific pattern. There are two main types of sequences: arithmetic and geometric.

An arithmetic sequence has a constant difference between consecutive terms. A geometric sequence has a constant ratio between consecutive terms.

To find the nth term of a sequence, you need to know the first term and either the common difference (for arithmetic sequences) or the common ratio (for geometric sequences).

Arithmetic Sequences

An arithmetic sequence is defined by its first term and common difference. The nth term of an arithmetic sequence can be found using the formula:

Arithmetic Sequence Formula:

aₙ = a₁ + (n - 1) × d

Where:

aₙ = nth term
a₁ = first term
d = common difference
n = term number

For example, if the first term (a₁) is 3 and the common difference (d) is 5, the 20th term would be calculated as:

a₂₀ = 3 + (20 - 1) × 5 = 3 + 95 = 98

The sequence would be: 3, 8, 13, 18, 23, ..., 98.

Geometric Sequences

A geometric sequence is defined by its first term and common ratio. The nth term of a geometric sequence can be found using the formula:

Geometric Sequence Formula:

aₙ = a₁ × r^(n - 1)

Where:

aₙ = nth term
a₁ = first term
r = common ratio
n = term number

For example, if the first term (a₁) is 2 and the common ratio (r) is 3, the 20th term would be calculated as:

a₂₀ = 2 × 3^(20 - 1) = 2 × 3¹⁹ ≈ 1,162,261,467

The sequence would be: 2, 6, 18, 54, 162, ..., 1,162,261,467.

Examples

Arithmetic Sequence Example

Given an arithmetic sequence with a₁ = 5 and d = 4, find the 20th term.

Using the formula:

a₂₀ = 5 + (20 - 1) × 4 = 5 + 76 = 81

The 20th term is 81.

Geometric Sequence Example

Given a geometric sequence with a₁ = 1 and r = 2, find the 20th term.

Using the formula:

a₂₀ = 1 × 2^(20 - 1) = 1 × 1,048,576 = 1,048,576

The 20th term is 1,048,576.

FAQ

What is the difference between arithmetic and geometric sequences?

An arithmetic sequence has a constant difference between consecutive terms, while a geometric sequence has a constant ratio between consecutive terms.

How do I find the nth term of a sequence?

For arithmetic sequences, use the formula aₙ = a₁ + (n - 1) × d. For geometric sequences, use aₙ = a₁ × r^(n - 1).

What if the sequence doesn't start at 1?

The formulas work regardless of where the sequence starts. Just use the given first term (a₁) and the appropriate difference or ratio.