Is 1, −1, 1, −1, 1, … a geometric progression? If so, what is the common ratio?

Step-by-step solution

Idea: A sequence is a GP when the ratio (term ÷ previous term) is the same for every pair of consecutive terms. That fixed ratio is the common ratio r.

1. Ratios of consecutive terms: −11 = −1, 1−1 = −1, −11 = −1, 1−1 = −1.1 mark
2. All the ratios are equal, so the sequence is a GP with common ratio −1 (each term is the previous term with its sign changed).1 mark

Check: First term × ratio repeatedly: 1, 1 × −1 = −1, … reproduces the sequence ✓.

Answer to write in the exam

Common mistakes that cost marks

• Saying it is not a GP because the terms do not grow. A GP can have a negative ratio; the terms then alternate in sign.
• Giving the ratio as 1 by ignoring the signs.
• Thinking the common difference is −2 and calling it an AP. The differences are −2, 2, −2, … so it is not an AP.

How this can come in the exam

Try one yourself

Is 3, −6, 12, −24, … a GP? If so, find the common ratio.

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