Sequence Property Counterexample Challenge

00:00 6 marks

Write a detailed answer that satisfies the following parts:

(a) Propose a property that a sequence might have, for example, ‘all terms are increasing’.

(b) Provide a specific counterexample sequence that demonstrates why this property does not hold universally, clearly explaining the failure.

Part a

4 minutes 3 marks

Question

Propose a sequence property. For example, state a property such as “every term is greater than the previous term.” Provide a concise statement of the proposed property.

Hint Hint

Hint

Think of a common sequence property, like monotonicity or boundedness.

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Complete Answer

An example property: “A sequence is increasing if for every n, \(a_{n+1} > a_n\).”

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Part b

4 minutes 3 marks

Question

Provide a counterexample sequence that disproves the universal validity of the property stated in part (a), and explain why it fails.

Hint Hint

Hint

Consider sequences that might appear to be increasing initially but fail to maintain the property.

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Complete Answer

A possible counterexample is the sequence defined by \(a_n = (-1)^n n\). This sequence does not satisfy the increasing property for all n because its terms alternate in sign and magnitude.

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