Sequences and Proof

This lesson covers proving properties of sequences, including convergence and divergence, and introduces proof by induction as awareness content for the Edexcel A-Level (9MA0) specification. Proof is a key mathematical skill and sequence-based proofs appear frequently in exams.

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Types of Proof Relevant to Sequences

Proof type	When to use
Direct proof	Showing a statement is true using logical steps from known results
Proof by contradiction	Assuming the opposite and reaching a contradiction
Proof by exhaustion	Checking all possible cases
Proof by induction	Proving a statement for all positive integers (A-Level awareness)

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Proving Sequence Properties Directly

Worked Example 1

Prove that the sum of the first n odd numbers is n².

The first n odd numbers are: 1, 3, 5, 7, ..., (2n - 1).

This is an AP with a = 1, d = 2, and the nth term is 2n - 1.

S(n) = n/2 x (first + last) = n/2 x (1 + (2n - 1)) = n/2 x 2n = n²

Worked Example 2