8.6 Patterns

**Conference tables (department Patterns and Formulae, January 2026).** Mr. Packer arranges square tables in a row. In Part One each table seats **3 people on each side and 1 person at each end**. (a) Draw or describe the arrangement for 1, 2, 3, 4 tables and complete a table of values. (b) Find an equation $M = \ldots$ for the number of people seated at $n$ tables. (c) Use it to find how many people can sit at 8 tables. (d) Verify by extending the table.

**General Packer formula (Part 3).** In a more general arrangement, $a$ people sit on each side of a table and $b$ people at each end. (a) Build the formula for $P$, the total people seated, in terms of $a$, $b$, $n$ (number of tables). (b) Check your formula against Part One ($a = 3, b = 1$) and Part Two ($a = 4, b = 2$). (c) Mr. Packer wants to use 4 tables that seat 5 on each side and 3 at each end. He thinks he can seat 40 people. Use your formula to determine whether he is correct.

**Linear sequence detective.** A sequence of pile-of-discs follows a linear pattern. The 4th term is 17 and the 10th term is 41. (a) Find the common difference. (b) Find the first term. (c) Write a formula for $T_n$. (d) Which term is equal to 101?

**Square-grid patterns.** Diagram $n$ contains an $n \times n$ grid of small squares: - Diagram 1: 1 small square - Diagram 2: 4 small squares - Diagram 3: 9 small squares - Diagram 4: 16 small squares (a) Write a formula for $T_n$ — the number of small squares in diagram $n$. (b) Find the difference between consecutive diagrams. What do you notice? (c) Prove algebraically that $T_{n+1} - T_n = 2n + 1$. (d) Use this to find $T_{50}$ from $T_{49}$ without squaring 50.

**Consecutive integers.** Three consecutive integers have a sum of 96. (a) Let the smallest be $n$. Write expressions for the other two and an equation for the sum. (b) Solve to find the integers. (c) Show that the sum of any three consecutive integers is always a multiple of 3. (d) Is the sum of four consecutive integers always a multiple of 4? Justify algebraically.

**Matchstick triangle pattern.** A linear pattern of triangles built from matchsticks: - 1 triangle: 3 sticks - 2 triangles: 5 sticks - 3 triangles: 7 sticks - 4 triangles: 9 sticks (a) Find $T_n$. (b) How many triangles can be made from 81 sticks? (c) Comment on the sticks left over.

**Triangular numbers.** $T_n = 1 + 2 + 3 + \ldots + n$. (a) Find $T_1, T_2, T_3, T_4, T_5$. (b) Find a closed-form formula for $T_n$. (c) Find $T_{50}$. (d) Prove (e.g. by pairing the sum from both ends) that $T_n = n(n+1)/2$.

**Fibonacci puzzle.** The Fibonacci sequence is defined by $F_1 = 1, F_2 = 1$, $F_{n+1} = F_n + F_{n-1}$. (a) Write the first 10 terms. (b) Find $F_{10}$. (c) Compute the ratio $F_{n+1}/F_n$ for $n = 5, 8, 10$. What do you notice as $n$ grows?

**Sum-formula investigation.** The sum of an arithmetic sequence with first term $a$ and common difference $d$ over $n$ terms is $S_n = \tfrac{n(2a + (n-1)d)}{2}$ (also $S_n = \tfrac{n(\text{first} + \text{last})}{2}$). (a) Verify the formula for $1 + 2 + 3 + \ldots + 10$. (b) Use the formula to find the sum of $5 + 8 + 11 + \ldots + 32$. (c) The first $n$ odd numbers ($1 + 3 + 5 + \ldots$) have sum $n^2$. Verify for $n = 5$.

**Repeating-decimal investigation.** Recurring decimals can be converted to fractions using the "method of 9s": $0.\overline{a} = a/9$, $0.\overline{ab} = ab/99$, etc. (a) Convert $0.\overline{3}$, $0.\overline{35}$, $0.\overline{017}$ to fractions in simplest form. (b) Show algebraically that $0.\overline{9} = 1$. (c) Use the sequence of partial sums to argue that $0.999\ldots$ approaches 1.

**Geometric pattern: paper folding.** A piece of paper is folded in half repeatedly. Each fold doubles the number of layers. (a) Find the number of layers after 1, 2, 3, 4, 5 folds. (b) Write a formula $L_n$ for the number of layers after $n$ folds. (c) Find $L_{10}$. (d) The Guinness world record for paper folds is 12. How many layers does that produce?

**Mixed sequence puzzle.** Consider the sequence 2, 6, 12, 20, 30, 42, … (a) Find the next two terms. (b) Show the differences and second differences. What does that suggest about the formula? (c) Notice that $T_n = n(n+1)$. Verify for $n = 1, 2, 3, 4, 5$. (d) Find $T_{20}$.