Unlocking Patterns: Arithmetic Sequences

 

Defining Arithmetic Sequences

An arithmetic sequence is a sequence of numbers in which each term after the first is obtained by adding a constant value, called the common difference ($d$), to the preceding term. The sequence can be expressed as:

• $a$, $a + d$, $a + 2d$, $a + 3d$, ..., $a + (n-1)d$

where $a$ is the first term, $d$ is the common difference, and $n$ represents the position of a term within the sequence.

Key Characteristics

• Linear Growth: Arithmetic sequences grow linearly, as depicted by their evenly spaced points when plotted on a graph.
• Uniform Difference: The difference between consecutive terms remains constant throughout the sequence.

 

Calculating Terms in an Arithmetic Sequence

The $n$th term of an arithmetic sequence can be calculated using the formula:

$$a_n = a + (n - 1)d$$

This formula helps determine any term's value based on its position ($n$) within the sequence.

 

Example: Finding a Specific Term

Given that the first term of a sequence ($a$) is 4 and the common difference ($d$) is 3, find the 5th term ($a_5$).

Substituting into the formula, we get $a_5 = 4 + (5 - 1)\times3 = 4 + 12 = 16$.