Unseen problem solving

Objective. Experiment, conjecture and prove in unfamiliar settings.

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Learn: the key idea

Unseen problem solving in Further Mathematics requires experimenting with small cases, forming a conjecture about a general pattern, and then proving that conjecture rigorously using an appropriate technique, rather than relying on a memorised method. A typical approach computes the first few terms or cases of a sequence or structure, looks for a pattern in the results, states a precise conjecture, and then verifies it using algebra, induction or another proof method suited to the pattern found. This experimental-then-rigorous cycle is central to tackling genuinely novel problems under exam conditions.

Worked example

By computing the first three triangular numbers' differences, conjecture a formula for the nth triangular number Tₙ and verify it for n=4.

1 $Triangular numbers: T₁=1, T₂=3, T₃=6; differences are 2 and 3, suggesting Tₙ adds n at each step$
2 $Conjecture: Tₙ = n(n+1)/2, based on the pattern of cumulative sums$
3 $Check for n=4: T₄ should be T₃+4 = 6+4 = 10$
4 $Using the formula: 4(5)/2 = 10, matching the computed value, supporting the conjecture$

Answer: Conjecture Tₙ = n(n+1)/2, verified for n=4 giving 10

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Common mistake. Students state a conjecture based on a pattern from a few cases and treat it as proven without any further justification. A pattern observed in several cases is only a conjecture until it is proved rigorously, typically by induction or direct algebraic argument.

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