Arithmetic patterns Knowledge-Based Study

Study notes

1. Understanding Arithmetic Patterns

   An arithmetic pattern (also known as an arithmetic sequence) is an ordered list of numbers where the difference between consecutive terms is always constant. This constant value added to each term to obtain the next term is called the common difference, often denoted by the letter $d$.

   Consider these examples:

   • Increasing pattern: $3, 7, 11, 15, 19, \dots$
     Here, each term increases by adding 4. The common difference is $d = +4$.
   • Decreasing pattern: $25, 20, 15, 10, 5, \dots$
     Here, each term decreases by subtracting 5. The common difference is $d = -5$.

   If the difference between consecutive terms varies at any point, the sequence is not an arithmetic pattern.

   Determining the Common Difference

   To determine whether a sequence is arithmetic and to find its common difference, subtract any term from the term that immediately follows it:

   $d = a_{n+1} - a_n$

   Let us test the sequence $8, 14, 20, 26, 32$:

   • $14 - 8 = 6$
   • $20 - 14 = 6$
   • $26 - 20 = 6$
   • $32 - 26 = 6$

   Because the difference between every adjacent pair is identically 6, the pattern is arithmetic with a common difference of $d = 6$.

   Non-example: Look at the sequence $2, 4, 8, 16, \dots$. The differences are $4 - 2 = 2$ and $8 - 4 = 4$. Because the differences are not constant, this is not an arithmetic pattern.

   Developing the Algebraic Formula

   Instead of continuing to add the common difference one step at a time, we can write an algebraic expression to determine any term directly based on its position, $n$.

   Let $a_1$ represent the first term and $d$ represent the common difference:

   • Term 1 ($n = 1$): $a_1$
   • Term 2 ($n = 2$): $a_1 + d$
   • Term 3 ($n = 3$): $a_1 + 2d$
   • Term 4 ($n = 4$): $a_1 + 3d$

   Notice that the number of times we add $d$ is always one less than the position number $n$. This gives us the general formula:

   $$a_n = a_1 + (n - 1)d$$

   For example, for the sequence $5, 9, 13, 17, \dots$:

   • First term ($a_1$) = 5
   • Common difference ($d$) = 4
   • General rule: $a_n = 5 + (n - 1)4 = 4n + 1$

   Solving Problems with Arithmetic Patterns

   Once you have the algebraic rule, you can calculate distant terms or check whether a specific number belongs to the sequence.

   Example 1: Finding a Distant Term

   Find the 40th term of the sequence $6, 11, 16, 21, \dots$.

   • Identify values: $a_1 = 6$, $d = 5$, $n = 40$
   • Apply formula: $a_{40} = 6 + (40 - 1) \times 5$
   • Simplify: $a_{40} = 6 + (39 \times 5) = 6 + 195 = 201$

   Example 2: Determining Membership

   Is the number 87 a term in the pattern $3, 7, 11, 15, \dots$?

   • General formula: $a_n = 3 + (n - 1) \times 4 = 4n - 1$
   • Set equal to 87: $4n - 1 = 87$
   • Solve for $n$: $4n = 88 \implies n = 22$

   Because $n = 22$ is a positive whole number, 87 is indeed the 22nd term of the pattern.