What is the common difference of the arithmetic sequence in which the 18th term minus the 14th term is 32?

The common difference of the arithmetic sequence is 8.

Understanding Arithmetic Sequences

An arithmetic sequence is a sequence of numbers such that the difference between any two consecutive terms is constant. This constant difference is called the common difference, often denoted by 'd'.

How to Calculate Common Difference

The general form of an arithmetic sequence is given by:

• a_n = a₁ + (n - 1)d

Where:

• a_n is the nth term
• a₁ is the first term
• n is the term number
• d is the common difference

When you have a term like a₁₈ and a term like a₁₄, their respective values can be expressed as:

• a₁₈ = a₁ + 17d
• a₁₄ = a₁ + 13d

The difference between a₁₈ and a₁₄ is thus:

• a₁₈ - a₁₄ = (a₁ + 17d) - (a₁ + 13d)
• a₁₈ - a₁₄ = 4d

Solving for the Common Difference

Given that a₁₈ - a₁₄ = 32, we can substitute this into the equation:

• 4d = 32

To solve for 'd', we simply divide both sides by 4:

• d = 32 / 4
• d = 8

Example

Let's verify with a simple example:
Suppose a₁ = 1 and the common difference, d = 8.

Then:

• a₁₄ = 1 + (14-1)8 = 1+138 = 105

• a₁₈ = 1 + (18-1)8 = 1+178 = 137

• a₁₈ - a₁₄ = 137 - 105 = 32

This confirms that when a₁₈ – a₁₄ = 32, the common difference is 8.