Patterns, Fibonacci and the golden ratio

What the board asks

Items in this area are mostly quick computations with the term and sum formulas, or word problems that hide a sequence.

• Arithmetic: nth term a + (n − 1)d. Sum (n ÷ 2)(2a + (n − 1)d).
• Geometric: nth term a × r^{n − 1}. Sum a(rⁿ − 1) ÷ (r − 1), for r not equal to 1.
• Fibonacci: F(n) = F(n − 1) + F(n − 2). The ratio of neighbours tends to φ ≈ 1.618.

Solved examples

Follow each one line by line. The steps stay the same when the numbers change, so cover the steps and try it yourself.

Example 1: Arithmetic sum in a word problem

Question: A hall has 12 seats in the first row and 4 more in each row after it. How many seats are in the first 10 rows?

1. Spot the pattern. "4 more in each row" is a fixed difference, and "how many in all" asks for a sum.
2. List what you know. a = 12, d = 4, n = 10.
3. Write the rule. S = (n ÷ 2)(2a + (n − 1)d).
4. Substitute. (10 ÷ 2)(2 × 12 + 9 × 4) = 5 × (24 + 36) = 5 × 60 = 300.

Answer: There are 300 seats.

Example 2: Geometric growth

Question: A pond has 200 fish, and the number triples every 4 years. How many fish are there after 12 years?

1. Spot the pattern. "Triples" means multiply by a fixed number, so it is geometric with r = 3.
2. Count the periods. 12 ÷ 4 = 3 triplings, not 12.
3. Apply the multiplier once per period. 200 × 3³ = 200 × 27 = 5400.

Answer: There are 5400 fish.

Example 3: Fibonacci and the golden ratio

Question: F(9) = 34 and F(10) = 55. Find F(12), then find F(12) ÷ F(11) to three decimal places.

1. Write the rule. F(n) = F(n − 1) + F(n − 2).
2. Build up. F(11) = 55 + 34 = 89. F(12) = 89 + 55 = 144.
3. Divide neighbours. 144 ÷ 89 = 1.6179…, which rounds to 1.618.
4. Look back. The result is close to φ ≈ 1.618, as neighbours in this sequence should be.

Answer: F(12) = 144, and the ratio is about 1.618.