Challenging Questions

Mathematics – Compulsory Part

Learning Unit: Arithmetic and geometric sequences and their summations

Question M0-07-09

Consider three sequences \(f(n)\), \(g(n)\) and \(h(n)\),

• \(f(n) = 1+x+x^2+x^3+\,…\,+x^{n-1}\)
• \(g(n) = 1-x+x^2-x^3+\,…\,+(-x)^{n-1}\)
• \(h(n) = 1+x^2+x^4+\,…\,+x^{2n-2}\)

1. Express \(f(n)\) as a rational function.
2. Express \(g(n)\) as a rational function, distinguishing between the cases for odd and even numbers.
3. Express \(h(n)\) as a rational function.
4. Hence show that \((1+x+x^2+x^3+\,…\,+x^{n-1}) \cdot (1-x+x^2-x^3+\,…\,+(-x)^{n-1}) = 1+x^2+x^4+\,…\,+x^{2n-2}\) when \(n\) is odd.

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