Number patterns

1. Linear Sequences (Arithmetic Patterns)

A linear sequence is a sequence of numbers where the difference between consecutive terms is constant. This constant value is called the common difference ($d$).

Key Formula

The general formula for the $n\text{-th}$ term ($T<em>n$) of a linear sequence is: $T</em>n = a + (n - 1)d$
Alternatively, expressed in simplified form:
$T_n = dn + c$

• $T_n$ represents the value of the term at position $n$.

• $n$ represents the position of the term in the sequence ($n \in {1, 2, 3, \dots}$).

• $a$ is the first term of the sequence ($T_1$).

• $d$ is the constant common difference ($d = T<em>2 - T</em>1 = T<em>3 - T</em>2$).

• $c$ is a constant value calculated as $c = a - d$

Worked Example 1: Finding the General Term

Consider the sequence: $5, 9, 13, 17, \dots$

1. Find the common difference ($d$):
   $d = 9 - 5 = 4$
   $d = 13 - 9 = 4$

2. Identify the first term ($a$):
   $a = 5$

3. Substitute into the general formula:
   $T<em>n = a + (n - 1)d$ $T</em>n = 5 + (n - 1)(4)$
   $T<em>n = 5 + 4n - 4$ $T</em>n = 4n + 1$

4. Find the $25\text{-th}$ term ($T<em>{25}$):
   $T</em>{25} = 4(25) + 1 = 101$

2. Quadratic Sequences

A quadratic sequence is a pattern where the second difference between consecutive terms is constant. The sequence of first differences forms a linear pattern.

Key Formula

The general term ($T<em>n$) for a quadratic sequence is given by: $T</em>n = an^2 + bn + c$
To determine the values of $a$, $b$, and $c$, use the relationship between the terms and differences:

1. Second Difference:
   $\text{Second Difference} = 2a$

2. First First Difference ($T<em>2 - T</em>1$):
   $T<em>2 - T</em>1 = 3a + b$

3. First Term ($T<em>1$):
   $T</em>1 = a + b + c$

Worked Example 2: Finding the General Formula for a Quadratic Sequence

Consider the sequence: $4, 9, 18, 31, 48, \dots$

1. Find the First Differences:

   • $9 - 4 = 5$

   • $18 - 9 = 9$

   • $31 - 18 = 13$

   • $48 - 31 = 17$
     First differences sequence: $5, 9, 13, 17$

2. Find the Second Differences:

   • $9 - 5 = 4$

   • $13 - 9 = 4$

   • $17 - 13 = 4$
     Constant second difference: $4$

3. Calculate Coefficients $a$, $b$, and $c$:

   • Find $a$:
     $2a = 4$
     $a = 2$

   • Find $b$:
     $3a + b = 5$
     $3(2) + b = 5$
     $6 + b = 5$
     $b = -1$

   • Find $c$:
     $a + b + c = 4$
     $2 + (-1) + c = 4$
     $1 + c = 4$
     $c = 3$

4. Write the General Formula:
   $T_n = 2n^2 - n + 3$

5. Verification (Check $T<em>3$):
   $T</em>3 = 2(3)^2 - (3) + 3 = 2(9) - 3 + 3 = 18$
   Matches the given sequence term $18$

3. Geometric Sequences (Exponential Patterns)

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio ($r$).

Key Formula

The general formula for the $n\text{-th}$ term ($T<em>n$) of a geometric sequence is: $T</em>n = a \cdot r^{n-1}$

• $T_n$ represents the value of the term at position $n$.

• $a$ is the first term of the sequence ($T_1$).

• $r$ is the constant common ratio ($r = \frac{T<em>2}{T</em>1} = \frac{T<em>3}{T</em>2}$).

Worked Example 3: Finding the General Term

Consider the sequence: $3, 6, 12, 24, \dots$

1. Find the common ratio ($r$):
   $r = \frac{6}{3} = 2$
   $r = \frac{12}{6} = 2$

2. Identify the first term ($a$):
   $a = 3$

3. Substitute into the general formula:
   $T_n = 3 \cdot 2^{n-1}$

4. Find the $8\text{-th}$ term ($T<em>8$):
   $T</em>8 = 3 \cdot 2^{8-1} = 3 \cdot 2^7 = 3 \cdot 128 = 384$

4. Cubic Sequences

A cubic sequence is a pattern where the third difference between consecutive terms is constant.

Key Formula

The general term ($T<em>n$) for a cubic sequence is given by: $T</em>n = an^3 + bn^2 + cn + d$
To determine the coefficients:

1. Third Difference:
   $\text{Third Difference} = 6a$

2. First Second Difference:
   $\text{First Second Difference} = 12a + 2b$

3. First First Difference ($T<em>2 - T</em>1$):
   $T<em>2 - T</em>1 = 7a + 3b + c$

4. First Term ($T<em>1$):
   $T</em>1 = a + b + c + d$

5. Special Sequences

1. Fibonacci Sequence:

   • A sequence where each term is the sum of the two preceding terms.

   • Sequence: $1, 1, 2, 3, 5, 8, 13, \dots$

   • Recursive relation: $T<em>n = T</em>{n-1} + T_{n-2}$ for $n \ge 3$

2. Triangular Numbers:

   • Numbers that can form an equilateral triangular grid.

   • Sequence: $1, 3, 6, 10, 15, \dots$

   • Formula: $T_n = \frac{n(n + 1)}{2}$

3. Square Numbers:

   • Sequence: $1, 4, 9, 16, 25, \dots$

   • Formula: $T_n = n^2$

4. Cube Numbers:

   • Sequence: $1, 8, 27, 64, 125, \dots$

   • Formula: $T_n = n^3$

6. Summary of Steps for Solving Number Patterns

1. Calculate First Differences: Check if the differences between consecutive terms are constant.

   • If constant, it is a linear sequence ($T_n = dn + c$).

2. Calculate Second Differences: If first differences are not constant, subtract consecutive first differences.

   • If constant, it is a quadratic sequence ($T_n = an^2 + bn + c$).

3. Calculate Ratio: Divide consecutive terms ($\frac{T<em>{n+1}}{T</em>n}$).

   • If constant, it is a geometric sequence ($T_n = a \cdot r^{n-1}$).

4. Calculate Third Differences: If second differences are not constant, subtract consecutive second differences.

   • If constant, it is a cubic sequence ($T_n = an^3 + bn^2 + cn + d$).

5. Check Special Sequences: Look for standard geometric patterns like square numbers ($n^2$), cube numbers ($n^3$), triangular numbers ($\frac{n(n+1)}{2}$), or recursive patterns (Fibonacci