Proof by Induction & Contradiction (DP IB Analysis & Approaches (AA): HL): Exam Questions

Prove by contradiction that $a^2-8b-7\ne 0$, for any $a,b\in \mathrm{\mathbb{Z}}$.

Prove by mathematical induction that, for $n\in {\mathrm{\mathbb{Z}}}^{+}$,

$1+2\left(\frac{1}{2}\right)+3{\left(\frac{1}{2}\right)}^2+4{\left(\frac{1}{2}\right)}^3+\dots +n{\left(\frac{1}{2}\right)}^{n-1}=4-\frac{n+2}{2^{n-1}}$

Prove by contradiction that the difference between a rational number and an irrational number is irrational.

Prove by mathematical induction that if $f\left(x\right)=x{\mathrm{e}}^{2x}$ then $f^{\left(n\right)}\left(x\right)=\left(2^{n}x+n{2}^{n-1}\right){\mathrm{e}}^{2x}$, for $n\in {\mathrm{\mathbb{Z}}}^{+}$.

Prove by mathematical induction that

${\left(\cos \theta -\mathrm{i}\sin \theta \right)}^n=\cos \left(n\theta \right)-\mathrm{i}\sin \left(n\theta \right)$

for all $n\in {\mathrm{\mathbb{Z}}}^{+}$.

Prove by contradiction that the equation $5x^4+15x^3-20x^2-4=0$ has no integer solutions.

Show that the derivative of $y=x{\mathrm{e}}^{-x}$ is

$\frac{\mathrm{d}y}{\mathrm{d}x}={\mathrm{e}}^{-x}\left(1-x\right)$.

Prove by mathematical induction that, for $n\in {\mathrm{\mathbb{Z}}}^{+}$,

$\frac{{\mathrm{d}}^{n}y}{\mathrm{d}{x}^n}={\mathrm{e}}^{-x}\left[{\left(-1\right)}^{n-1}n+{\left(-1\right)}^{n}x\right]$

Prove by mathematical induction that ${\left[r\left(\cos \theta -\mathrm{i}\sin \theta \right)\right]}^n=r^n\left[\cos \left(n\theta \right)-\mathrm{i}\sin \left(n\theta \right)\right]$ for all $n\in {\mathrm{\mathbb{Z}}}^{+}$.

Prove by mathematical induction that $2^{n+2}+3^{3n}$ is divisible by 5 for all $n\in \mathrm{\mathbb{N}}$.

Prove by contradiction that there are infinitely many prime numbers.

Prove by mathematical induction that

$\sum _{r=1}^{n}r\left(r^2-6\right)=\frac{1}{4}n\left(n+1\right)\left(n+4\right)\left(n-3\right)$

for $n\in {\mathrm{\mathbb{Z}}}^{+}$.