Triangular, Square, Cube and Fibonacci Sequences

• Square Number sequences represent the area of a square ($n \times n$).

• The first five terms are 1, 4, 9, 16, 25.

• The gap between consecutive terms increases by the next odd number ($+3, +5, +7$), giving them a constant second difference of $2$.

• Cube Number sequences represent the volume of a cube ($n \times n \times n$).

• The first five terms are 1, 8, 27, 64, 125.

• The number $64$ is unique as it is a shared term: it is both a square number ($8^2$) and a cube number ($4^3$). You must also recognise that $10^3 = 1000$.

Using the nth Term Formulas

The nth term is a position-to-term rule that calculates a term directly from its position ($n$) without needing to know the previous term. For square numbers, the formula is $n^2$, and for cube numbers, it is $n^3$.

The triangular nth term formula is slightly more complex:

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

Often, sequences are "shifted" versions of these standard rules. For the sequence $4, 7, 12, 19...$, comparing it to the standard square numbers ($1, 4, 9, 16...$) reveals that every term is $3$ larger, so the formula is $n^2 + 3$.

Worked Example: Testing a Triangular Number (Higher Tier)

Show that 120 is a triangular number and find its position in the sequence.

Step 1: Set the triangular nth term formula equal to the target number.

• $\frac{n(n+1)}{2} = 120$

Step 2: Multiply by 2 to remove the fraction.

• $n(n+1) = 240$
• $n^2 + n = 240$

Step 3: Rearrange into a quadratic equation equal to zero and factorise.

• $n^2 + n - 240 = 0$
• $(n+16)(n-15) = 0$

Step 4: Interpret the solutions.

• $n = -16$ or $n = 15$
• Since a position ($n$) must be a positive integer, $n = 15$. Therefore, 120 is the 15th triangular number.

Fibonacci-Type Sequences

The spiral of a snail shell and the arrangement of seeds in a sunflower both follow a famous mathematical pattern: the Fibonacci sequence. A Fibonacci Sequence uses a term-to-term rule where each term is the sum of the two preceding terms.

• The standard Fibonacci sequence starts: 0, 1, 1, 2, 3, 5, 8, 13, 21...
• Every Fibonacci-type sequence requires two seed values to begin.
• Any two starting numbers can generate a valid sequence. For example, using $3$ and $7$ gives: $3, 7, 10, 17, 27...$
• As the sequence progresses, the ratio of consecutive terms ($\frac{a_{n+1}}{a_n}$) gets closer and closer to the Golden Ratio (approximately ).

Worked Example: Reversing a Fibonacci Sequence

The 3rd and 4th terms of a Fibonacci-type sequence are 7 and 11. Find the 1st and 2nd terms.

Step 1: Use the rule backwards to find the 2nd term (Term 4 - Term 3).

• $11 - 7 = 4$

Step 2: Use the rule backwards to find the 1st term (Term 3 - Term 2).

• $7 - 4 = 3$

Step 3: State the final sequence.

• The sequence begins: 3, 4, 7, 11.

Algebraic Fibonacci Sequences (Higher Tier)

Fibonacci sequences are often presented algebraically in Edexcel exams. If the first two seed values are $a$ and $b$, the first six terms of the sequence will always follow this pattern:

• 1st term: $a$
• 2nd term: $b$
• 3rd term: $a + b$
• 4th term: $a + 2b$
• 5th term: $2a + 3b$

Worked Example: Finding Unknown Seed Values

The 3rd term of an algebraic Fibonacci sequence is 7 and the 6th term is 29. Find the values of the first term ($a$) and the second term ($b$).

Step 1: Form simultaneous equations using the standard algebraic pattern.

• $a + b = 7$
• $3a + 5b = 29$

Step 2: Multiply the first equation by 3 to match the $a$ coefficients.

• $3(a + b) = 3(7)$
• $3a + 3b = 21$

Step 3: Subtract this from the second equation to eliminate $a$.

• $(3a + 5b) - (3a + 3b) = 29 - 21$
• $2b = 8$

Step 4: Substitute $b = 4$ back into the first equation to find $a$.

• $a + 4 = 7$
• $a = 3$

Step 5: State the final values.

• $a = 3$, $b = 4$