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Algebra · Sequences

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Geometric, Fibonacci and other special sequences

Why geometric sequences multiply where arithmetic ones add, how Fibonacci-type rules turn into algebra, the named sequences worth recognising on sight, and subscript notation for term-to-term rules.

Algebra · Sequences

Geometric, Fibonacci and other special sequences

Why geometric sequences multiply where arithmetic ones add, how Fibonacci-type rules turn into algebra, the named sequences worth recognising on sight, and subscript notation for term-to-term rules.

Why it works

Not every sequence climbs by adding. The exam's favourite "other" rules:

Geometric sequences MULTIPLY by a constant ratio. 3,6,12,243, 6, 12, 24 doubles each time — the next terms are 48,9648, 96. Spot the ratio by dividing neighbours: 63=126=2\frac{6}{3} = \frac{12}{6} = 2. Continuing a geometric sequence by adding the first gap (24 + 3 = 27) is the classic trap — always test whether neighbours differ by a sum or a factor.

The ratio can be a surd. 3,  3,  33,  9,…\sqrt{3},\; 3,\; 3\sqrt{3},\; 9, \ldots multiplies by 3\sqrt{3} each step (3÷3=33 \div \sqrt{3} = \sqrt{3} after rationalising). Next term: 9×3=939 \times \sqrt{3} = 9\sqrt{3}. Surd-ratio chains are pure Paper 1 fodder — simplify as you go.

Fibonacci-type sequences add the previous TWO terms. 2,3,5,8,13→21,342, 3, 5, 8, 13 \to 21, 34. The exam's Higher twist makes it algebra: if the first two terms are aa and bb, the sequence runs

a,  b,  a+b,  a+2b,  2a+3b,  …a,\; b,\; a + b,\; a + 2b,\; 2a + 3b,\; \ldots

Given values of later terms, that's simultaneous equations: if the 4th term is 10 and the 5th is 17, then a+2b=10a + 2b = 10 and 2a+3b=172a + 3b = 17, giving b=3b = 3, a=4a = 4 — sequence 4,3,7,10,174, 3, 7, 10, 17. ✓

Know the named sequences on sight:
  • square numbers 1,4,9,16,25,…1, 4, 9, 16, 25, \ldots
  • cube numbers 1,8,27,64,…1, 8, 27, 64, \ldots
  • triangular numbers 1,3,6,10,15,21,…1, 3, 6, 10, 15, 21, \ldots — each adds one
more than the last (+2,+3,+4,…+2, +3, +4, \ldots), the totals of stacked rows;
  • powers of 2: 1,2,4,8,16,…1, 2, 4, 8, 16, \ldots
Subscript notation is just a term-to-term rule in symbols. xn+1=3xn−2x_{n+1} = 3x_n - 2 says "next == 3 × current −2- 2". From x1=4x_1 = 4: x2=3(4)−2=10x_2 = 3(4) - 2 = 10, x3=3(10)−2=28x_3 = 3(10) - 2 = 28. Read the subscript as a position label, not algebra to expand.