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Ecolint Campus des NationsMathématiques
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Problem-solving Pack

Ecolint Campus des NationsMathematics
Year 8 · 8.6 Patterns

Problem-solving Pack

Name: _________________________________
Date: _________________ Class: ___________

These problems are designed to challenge you. Read each question carefully. Show all your reasoning — a correct answer without working receives no credit.

1Problem 1 of 12
Conference tables (CdN 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=…M = \ldots for the number of people seated at nn tables.
(c) Use it to find how many people can sit at 8 tables.
(d) Verify by extending the table.

Working space

2Problem 2 of 12
General Packer formula (Part 3). In a more general arrangement, aa people sit on each side of a table and bb people at each end.

(a) Build the formula for PP, the total people seated, in terms of aa, bb, nn (number of tables).
(b) Check your formula against Part One (a=3,b=1a = 3, b = 1) and Part Two (a=4,b=2a = 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.

Working space

3Problem 3 of 12
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 TnT_n.
(d) Which term is equal to 101?

Working space

4Problem 4 of 12
Square-grid patterns. Diagram nn contains an n×nn \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 TnT_n — the number of small squares in diagram nn.
(b) Find the difference between consecutive diagrams. What do you notice?
(c) Prove algebraically that Tn+1−Tn=2n+1T_{n+1} - T_n = 2n + 1.
(d) Use this to find T50T_{50} from T49T_{49} without squaring 50.

Working space

5Problem 5 of 12
Consecutive integers. Three consecutive integers have a sum of 96.

(a) Let the smallest be nn. 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.

Working space

6Problem 6 of 12
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 TnT_n.
(b) How many triangles can be made from 81 sticks?
(c) Comment on the sticks left over.

Working space

7Problem 7 of 12
Triangular numbers. Tn=1+2+3+…+nT_n = 1 + 2 + 3 + \ldots + n.

(a) Find T1,T2,T3,T4,T5T_1, T_2, T_3, T_4, T_5.
(b) Find a closed-form formula for TnT_n.
(c) Find T50T_{50}.
(d) Prove (e.g. by pairing the sum from both ends) that Tn=n(n+1)/2T_n = n(n+1)/2.

Working space

8Problem 8 of 12
Fibonacci puzzle. The Fibonacci sequence is defined by F1=1,F2=1F_1 = 1, F_2 = 1, Fn+1=Fn+Fn−1F_{n+1} = F_n + F_{n-1}.

(a) Write the first 10 terms.
(b) Find F10F_{10}.
(c) Compute the ratio Fn+1/FnF_{n+1}/F_n for n=5,8,10n = 5, 8, 10. What do you notice as nn grows?

Working space

9Problem 9 of 12
Sum-formula investigation. The sum of an arithmetic sequence with first term aa and common difference dd over nn terms is Sn=n(2a+(n−1)d)2S_n = \tfrac{n(2a + (n-1)d)}{2} (also Sn=n(first+last)2S_n = \tfrac{n(\text{first} + \text{last})}{2}).

(a) Verify the formula for 1+2+3+…+101 + 2 + 3 + \ldots + 10.
(b) Use the formula to find the sum of 5+8+11+…+325 + 8 + 11 + \ldots + 32.
(c) The first nn odd numbers (1+3+5+…1 + 3 + 5 + \ldots) have sum n2n^2. Verify for n=5n = 5.

Working space

10Problem 10 of 12
Repeating-decimal investigation. Recurring decimals can be converted to fractions using the "method of 9s":

0.a‾=a/90.\overline{a} = a/9, 0.ab‾=ab/990.\overline{ab} = ab/99, etc.

(a) Convert 0.3‾0.\overline{3}, 0.35‾0.\overline{35}, 0.017‾0.\overline{017} to fractions in simplest form.
(b) Show algebraically that 0.9‾=10.\overline{9} = 1.
(c) Use the sequence of partial sums to argue that 0.999…0.999\ldots approaches 1.

Working space

11Problem 11 of 12
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 LnL_n for the number of layers after nn folds.
(c) Find L10L_{10}.
(d) The Guinness world record for paper folds is 12. How many layers does that produce?

Working space

12Problem 12 of 12
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 Tn=n(n+1)T_n = n(n+1). Verify for n=1,2,3,4,5n = 1, 2, 3, 4, 5.
(d) Find T20T_{20}.

Working space