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

Prove by mathematical induction that  $9^{2n}-1,n\in \mathrm{\mathbb{Z}},n\ge 1$ is divisible by 16.

Prove by contradiction that $\sqrt{10}$ is irrational.

Use mathematical induction to prove that the $n$th derivative of the function $f\left(x\right)=\frac{5}{x}$ is  given by

 $\frac{5{\left(-1\right)}^{n}n!}{x^{\left(n+1\right)}}$ 

for all integers, $n$, where $n\ge 1$.

Use proof by contradiction to prove that $a^2-8b-11\ne 0$ if $a,b\in \mathrm{\mathbb{Z}}$.

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

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

Use a contradiction to prove that the difference between a rational number and an irrational number is irrational.

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

Prove by mathematical induction that

${\left(\cos \theta -\mathrm{i}\sin \theta \right)}^n=\cos \left(n\theta \right)-\mathrm{i}\sin \left(n\theta \right),\mathrm{for}\mathrm{all}n\in {\mathrm{\mathbb{Z}}}^{+}$

Prove that $2^{n+2}+3^{3n}$ is divisible by 5 for $n\in \mathrm{\mathbb{Z}},n\ge 0$.

Prove that there are an infinite number of prime numbers.

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

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

$\frac{dy}{dx}=e^{-x}\left(1-x\right)$.

Prove, by mathematical induction, that for $n\ge 1,$

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

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

Prove by 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 all positive integer values of $n$.