Numerical patterns Knowledge-Based Study

Study notes

1. What is a Numerical Pattern?

   A numerical pattern (or number sequence) is an ordered list of numbers that follows a specific mathematical rule. Each number in the pattern is called a term.

   Understanding patterns requires distinguishing between two key concepts:

   • Term position ($n$): The position or index of the term in the list (for example, 1st term, 2nd term, 3rd term, etc.). Position numbers are always positive integers: $n = 1, 2, 3, \dots$
   • Term value: The actual number that appears at that position.

   For example, consider the sequence: 4, 7, 10, 13, 16, …

   • When $n = 1$, the term value is 4.
   • When $n = 2$, the term value is 7.
   • When $n = 3$, the term value is 10.

   Identifying Linear Patterns

   A pattern is called linear (or an arithmetic sequence) when the difference between any two consecutive terms is constant. This fixed value is known as the common difference ($d$).

   To calculate the common difference, subtract any term from the term that immediately follows it:

   $$d = \text{Term}_{n+1} - \text{Term}_n$$

   Increasing vs. Decreasing Patterns

   • Increasing pattern: The common difference is positive. For example, in 6, 11, 16, 21, …, the difference is $11 - 6 = +5$.
   • Decreasing pattern: The common difference is negative. For example, in 28, 24, 20, 16, …, the difference is $24 - 28 = -4$.

   If the difference between consecutive terms is not constant, the pattern is non-linear.

   The General Rule Formula

   Rather than continuing a pattern step by step to find distant terms, we can determine an algebraic expression for the $n$-th term (denoted as $T_n$). For any linear sequence, the general expression takes the form:

   $$T_n = dn + c$$

   where $d$ is the common difference and $c$ is an adjusted constant value.

   Step-by-Step Procedure

   Consider the pattern: 3, 8, 13, 18, 23, …

   1. Find the common difference ($d$):
      Subtract the first term from the second term: $8 - 3 = 5$. Since the step is $+5$, the rule begins with $5n$.
   2. Find the constant ($c$):
      Substitute $n = 1$ into $5n$: $5 \times 1 = 5$.
      The actual first term is 3. To go from 5 to 3, subtract 2.
      Therefore, $c = 3 - 5 = -2$.
   3. Combine to state the rule:
      $$T_n = 5n - 2$$

   Applications of the nth Term

   Once you have the algebraic rule for a pattern, you can evaluate any term directly or determine whether a specific number belongs to the sequence.

   Example 1: Finding a Specified Term

   Using the rule $T_n = 5n - 2$, find the 40th term ($n = 40$):

   $$T_{40} = 5(40) - 2 = 200 - 2 = 198$$

   Example 2: Checking Membership in a Sequence

   To determine if a given number is a term in the sequence, set $T_n$ equal to that number and solve for $n$. If $n$ is a positive whole number, the value belongs to the sequence.

   Is 173 a term in the pattern $T_n = 5n - 2$?

   $$5n - 2 = 173$$

   $$5n = 175$$

   $$n = 35$$

   Since 35 is a positive integer, 173 is indeed the 35th term of the pattern.

   Is 90 a term in the same pattern?

   $$5n - 2 = 90 \implies 5n = 92 \implies n = 18.4$$

   Because 18.4 is not an integer, 90 is not a term in this pattern.