Number sequences and patterns

A sequence is an ordered list of numbers built by a rule. The rule can be plain ("add $3$ each time") or sneaky ("each term is the sum of the previous two"). Spotting the rule is one of the most useful skills in mathematics , and one of the most enjoyable puzzles.

Concept

Arithmetic sequences. Each term is obtained by adding a fixed number, the common difference $d$, to the previous term. Examples:

• $3, 7, 11, 15, \dots$ with $d = 4$.
• $20, 17, 14, 11, \dots$ with $d = -3$.

General term: $a_n = a_1 + (n-1)d$. Sum of the first $n$ terms: $S_n = \dfrac{n}{2}(a_1 + a_n)$.

Geometric sequences. Each term is obtained by multiplying the previous one by a fixed common ratio $r$. Examples:

• $2, 6, 18, 54, \dots$ with $r = 3$.
• $80, 40, 20, 10, \dots$ with $r = \tfrac{1}{2}$.

General term: $a_n = a_1 \cdot r^{n-1}$. These show exponential growth or decay.

Triangular numbers. $1, 3, 6, 10, 15, \dots$ , counts of dots in growing triangles. Formula: $T_n = \dfrac{n(n+1)}{2}$.

Square numbers. $1, 4, 9, 16, 25, \dots$ , $S_n = n^2$.

Fibonacci sequence. $1, 1, 2, 3, 5, 8, 13, 21, 34, 55, \dots$ Each term is the sum of the two before it. Discovered (in Europe) by Fibonacci in 1202 in a problem about rabbits, this sequence later showed up in pinecones, sunflower spirals, and family trees of honeybees. The ratio of consecutive terms approaches the golden ratio $\varphi = \dfrac{1 + \sqrt{5}}{2} \approx 1.618$.

Pascal's triangle. Each entry is the sum of the two above it.

$\begin{array}{c}1 \\ 1\;1 \\ 1\;2\;1 \\ 1\;3\;3\;1 \\ 1\;4\;6\;4\;1 \\ 1\;5\;10\;10\;5\;1\end{array}$

This single triangle contains the powers of $11$ ($11^4 = 14641$), the Fibonacci sequence (sum of "shallow diagonals"), and the binomial coefficients you will meet in algebra.

Recognising the rule. When given a sequence:

1. Compute the differences between consecutive terms. Constant? Arithmetic.
2. Compute the ratios. Constant? Geometric.
3. Compute second differences. Constant? Quadratic (like square numbers).
4. Try summing two previous terms (Fibonacci-style).

Worked examples

Example 1. Find the $20$-th term of $5, 8, 11, 14, \dots$.

• Common difference $d = 3$, first term $a_1 = 5$.
• $a_{20} = 5 + 19 \cdot 3 = 5 + 57 = 62$.

Example 2. Find the sum of the first $50$ terms of $1, 3, 5, 7, \dots$.

• This is arithmetic with $d = 2$, $a_1 = 1$.
• $a_{50} = 1 + 49 \cdot 2 = 99$.
• $S_{50} = \dfrac{50}{2}(1 + 99) = 25 \cdot 100 = 2500$.

Example 3. Find the $7$-th term of geometric sequence $3, 6, 12, 24, \dots$.

• $r = 2$, $a_1 = 3$.
• $a_7 = 3 \cdot 2^6 = 3 \cdot 64 = 192$.

Example 4. Find the $10$-th Fibonacci number, starting $1, 1, 2, 3, \dots$.

• Continue: $5, 8, 13, 21, 34, 55$.
• Tenth term: $55$.

Try it yourself

1. Find the $25$-th term of $2, 5, 8, 11, \dots$.
2. Find the sum of the first $30$ positive even numbers.
3. Find the next two terms of $1, 4, 9, 16, 25, \dots$.
4. Find the $8$-th term of geometric sequence $5, 15, 45, \dots$.
5. List the first $12$ Fibonacci numbers starting from $1, 1$.
6. Spot the rule and give the next two terms: $2, 6, 12, 20, 30, \dots$.
7. The first three terms of an arithmetic sequence are $7, 12, 17$. Find $a_{50}$.
8. Compute the ratio $F_{10}/F_{9}$ (where $F$ are Fibonacci) and compare with the golden ratio $\approx 1.618$.

Activity / Insight

Sunflower hunt. Look up a high-resolution picture of a sunflower head. Count the number of spirals winding clockwise and the number winding anticlockwise. You will find Fibonacci numbers , often $34$ and $55$, sometimes $55$ and $89$. This is the most-photographed example of how an abstract sequence shows up in nature.