Common Number Patterns and Fibonacci Sequences

Generating Terms Using Formulas

To find any number in these sequences without writing out the whole list, we use a position-to-term rule (nth term). This is a formula that allows calculation of any term using its position, $n$.

To generate the first five terms, substitute $n = 1, 2, 3, 4, 5$ into the respective formulas:

1. Square Numbers: Formula is $n^2$

• Step 1: Substitute $n=1 \rightarrow 1^2 = 1$
• Step 2: Substitute $n=2 \rightarrow 2^2 = 4$
• Step 3: Substitute $n=3 \rightarrow 3^2 = 9$
• Step 4: Substitute $n=4 \rightarrow 4^2 = 16$
• Step 5: Substitute $n=5 \rightarrow 5^2 = 25$
• The first five terms are 1, 4, 9, 16, 25.

2. Cube Numbers: Formula is $n^3$

• Step 1: Substitute $n=1 \rightarrow 1^3 = 1$
• Step 2: Substitute $n=2 \rightarrow 2^3 = 8$
• Step 3: Substitute $n=3 \rightarrow 3^3 = 27$
• Step 4: Substitute $n=4 \rightarrow 4^3 = 64$
• Step 5: Substitute $n=5 \rightarrow 5^3 = 125$
• The first five terms are 1, 8, 27, 64, 125.

3. Triangular Numbers: Formula is $\frac{n(n+1)}{2}$

• Step 1: Substitute $n=1 \rightarrow \frac{1(1+1)}{2} = 1$
• Step 2: Substitute $n=2 \rightarrow \frac{2(2+1)}{2} = 3$
• Step 3: Substitute $n=3 \rightarrow \frac{3(3+1)}{2} = 6$
• Step 4: Substitute $n=4 \rightarrow \frac{4(4+1)}{2} = 10$
• Step 5: Substitute $n=5 \rightarrow \frac{5(5+1)}{2} = 15$
• The first five terms are 1, 3, 6, 10, 15.

Calculating Specific Distant Terms

Examiners frequently ask you to calculate a distant term, such as the 12th or 22nd term. You simply substitute the required position number into the formula.

Worked Example 1: Calculate the 12th square number.

• Step 1: Identify the formula: $n^2$
• Step 2: Substitute $n = 12$: $12^2$
• Step 3: Calculate the final answer: $144$

Worked Example 2: Calculate the 22nd triangular number.

• Step 1: Identify the formula: $\frac{n(n+1)}{2}$
• Step 2: Substitute $n = 22$: $\frac{22(22+1)}{2}$
• Step 3: Simplify the numerator: $\frac{22 \times 23}{2}$
• Step 4: Calculate the final answer: $11 \times 23 = 253$

Understanding Fibonacci-Type Sequences

A Fibonacci-type sequence is a pattern of numbers generated by an additive property where each term is the sum of the two preceding terms. Unlike linear or quadratic sequences, Fibonacci sequences do NOT have a constant first or second difference.

The standard Fibonacci sequence begins 1, 1, 2, 3, 5, 8, 13. The term-to-term rule defining how to get from one term to the next is simply "add the two previous terms". However, Fibonacci-type sequences can start with any two integers, including negative numbers or decimals.

Worked Example 3: Generating a sequence with negative starting integers.

Generate the first 5 terms of a Fibonacci-type sequence starting with $-3$ and $5$.

• Step 1: State the 1st and 2nd terms: $-3, 5$
• Step 2: Add the 1st and 2nd terms to find the 3rd term: $-3 + 5 = 2$
• Step 3: Add the 2nd and 3rd terms to find the 4th term: $5 + 2 = 7$
• Step 4: Add the 3rd and 4th terms to find the 5th term: $2 + 7 = 9$
• The sequence is $-3, 5, 2, 7, 9$.

Finding Missing Middle Terms

You can also work backwards or use algebra to find missing terms in a Fibonacci-type sequence. If given a sequence with a gap, represent the unknown term with a letter.

Worked Example 4: Find the missing terms in the sequence $4, \_\_, 15, \_\_$

• Step 1: Let the missing 2nd term be $x$.
• Step 2: Apply the rule that the 1st and 2nd terms sum to the 3rd term: $4 + x = 15$
• Step 3: Solve for $x$: $x = 11$
• Step 4: Add the 2nd and 3rd terms to find the 4th term: $11 + 15 = 26$
• The complete sequence is $4, 11, 15, 26$.