Arithmetic and Geometric Progression

 

Defining Arithmetic Progression

An arithmetic progression is a progression 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 progression 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 progression.

Key Characteristics

• Linear Growth: Arithmetic progressions 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 progression.

 

Calculating Terms in an Arithmetic Progression

The $n$th term of an arithmetic progression 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 progression.

 

Example: Finding a Specific Term

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

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

 

 

 

 

 

Geometric progressions

A geometric progression is a progression where each term is found by multiplying the previous term by a constant number called the common ratio, $r$.

General Formula: $$a_n = a_1 \dot r^(n-1)$$

 

where:

• $a_1$ is the first term,
• $r$ is the common ratio,
• $n$ is the position of the term 

 

Example:

Find the first 5 terms of the geometric progression where $a_1 = 3$ and $r = 2$

Solution: $$3, 6, 12, 24, 48, ...$$