Mathematics AA: MATHEMATICAL INDUCTION Topical Past Paper Questions

[Maximum mark:27] This question asks you to explore self-composite linear functions, of the form $f(x)=mx+c$, for varying values of $m$. A function composed with itself is called a self-composite function. For a function $ f $, the function composition with itself is given by $ (…

[Maximum mark: 20] (a) Let $ f(x)=(1-ax)^{-\frac{1}{2}} $, where $ ax [8] (b) By using part (a) or otherwise, show that the Maclaurin series for $ f(x)=(1-ax)^{-\frac{1}{2}} $ up to and including the $ x^2 $ term is $ 1+\frac{1}{2}ax+\frac{3}{8}a^2 x^2 $. [2] (c) Hence, show that…

[Maximum mark: 7] Prove by mathematical induction that $5^{2n} - 2^{3n}$ is divisible by 17 for all $n \in \mathbb{Z}^{+}$.