1.15 Proof by induction, contradiction and counterexample questions for IB Maths AA

Show that $\arctan p-\arctan q=\arctan \left({\displaystyle \frac{p-q}{1+pq}}\right)$, for $p,q\ge 0$.

Hence show that $\arctan (n+2)-\arctan n=\arctan \left({\displaystyle \frac{2}{(n+1{)}^2}}\right)$, for $n\ge 0$.

Show that $S_2={\displaystyle \frac{\pi }{2}}$.

Prove by mathematical induction that $S_n=\arctan n+\arctan (n+1)-{\displaystyle \frac{\pi }{4}}$, for $n\in {\mathbb{Z}}^{+}$.

Find ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }{S}_n}$.

Use your GDC to find the values of $n$, where $1\le n\le 12$, for which the inequality is true.

Prove by mathematical induction that $2^n>n^3$ for all integers $n\ge 10$.

Use a double angle identity to show that $T_2(x)=2x^2-1$.

By expanding $\cos ((n+1)\theta )$ and $\cos ((n-1)\theta )$, show that $\cos ((n+1)\theta )+\cos ((n-1)\theta )\equiv 2\cos \theta \cos n\theta$.

Hence show that $T_{n+1}(x)=2x{T}_n(x)-T_{n-1}(x)$, for $n\in {\mathbb{Z}}^{+}$.

Hence find $T_3(x)$ and $T_4(x)$.

Use your GDC to sketch the graph of $y=T_4(x)$ for $-1\le x\le 1$, and write down the range of $T_4$ on this interval.

Write down the leading coefficient of $T_n(x)$ for $n=1,2,3,4$, and hence suggest an expression for the leading coefficient of $T_n(x)$, for $n\in {\mathbb{Z}}^{+}$.

Prove by mathematical induction that, for $n\in {\mathbb{Z}}^{+}$, $T_n(x)$ is a polynomial of degree $n$ with leading coefficient $2^{n-1}$.

Show that the $n$ roots of $T_n(x)=0$ are $x=\cos \left({\displaystyle \frac{(2k-1)\pi }{2n}}\right)$, for $k=1,2,\dots ,n$.

Hence find the exact value of $\cos {\displaystyle \frac{\pi }{8}}$.

By considering the product of the roots of $T_4(x)=0$, find the exact value of $\cos {\displaystyle \frac{\pi }{8}}\cos {\displaystyle \frac{3\pi }{8}}$.

Find the value of $k$ for which $\mathbf{a}$ and $\mathbf{b}$ are parallel.

For this value of $k$, write down the value of ${\displaystyle \frac{\mathrm{\mid }\mathbf{b}\mathrm{\mid }}{\mathrm{\mid }\mathbf{a}\mathrm{\mid }}}$.

Let $\mathbf{u}$ and $\mathbf{v}$ be any two non-zero vectors that are not parallel. Prove by contradiction that $\mathbf{u}+\mathbf{v}$ and $\mathbf{u}-\mathbf{v}$ are not parallel.

Prove by mathematical induction that

An investment grows by $3\mathrm{\%}$ each year, so after $40$ years its value has been multiplied by ${1.03}^{40}$.

Use part (a) to find a lower bound for ${1.03}^{40}$, and find the percentage error when this lower bound is used as an estimate of ${1.03}^{40}$.

Using the result $\mathbf{a}\cdot \mathbf{b}=\mathrm{\mid }\mathbf{a}\mathrm{\mid }\mathrm{\mid }\mathbf{b}\mathrm{\mid }\cos \theta$, where $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{b}$, show that $\mathrm{\mid }\mathbf{a}\cdot \mathbf{b}\mathrm{\mid }\le \mathrm{\mid }\mathbf{a}\mathrm{\mid }\mathrm{\mid }\mathbf{b}\mathrm{\mid }$, and state the condition on $\mathbf{a}$ and $\mathbf{b}$ for which equality holds.

Let $\mathbf{a}=\left(\begin{array}{c}2\\ 3\\ 6\end{array}\right)$ and $\mathbf{b}=\left(\begin{array}{c}x\\ y\\ z\end{array}\right)$, where $x^2+y^2+z^2=1$.

Show that $2x+3y+6z\le 7$.

Find the values of $x$, $y$ and $z$ for which $2x+3y+6z=7$.

The proof in part (a) assumes that the angle $\theta$ exists, which itself relies on ${\displaystyle \frac{\mathbf{a}\cdot \mathbf{b}}{\mathrm{\mid }\mathbf{a}\mathrm{\mid }\mathrm{\mid }\mathbf{b}\mathrm{\mid }}}$ lying between $-1$ and $1$. In the rest of the question, you will prove the inequality without using angles.

Let $f(t)=\mathrm{\mid }\mathbf{a}+t\mathbf{b}{\mathrm{\mid }}^2$, where $t\in \mathbb{R}$.

Show that $f(t)=\mathrm{\mid }\mathbf{b}{\mathrm{\mid }}^2{t}^2+2(\mathbf{a}\cdot \mathbf{b})t+\mathrm{\mid }\mathbf{a}{\mathrm{\mid }}^2$.

Explain why $f(t)\ge 0$ for all $t\in \mathbb{R}$, and hence show that $(\mathbf{a}\cdot \mathbf{b}{)}^2\le \mathrm{\mid }\mathbf{a}{\mathrm{\mid }}^2\mathrm{\mid }\mathbf{b}{\mathrm{\mid }}^2$.

Show that if $(\mathbf{a}\cdot \mathbf{b}{)}^2=\mathrm{\mid }\mathbf{a}{\mathrm{\mid }}^2\mathrm{\mid }\mathbf{b}{\mathrm{\mid }}^2$, then $\mathbf{a}$ and $\mathbf{b}$ are parallel.

Use the result of part (d) to prove that $\mathrm{\mid }\mathbf{a}+\mathbf{b}\mathrm{\mid }\le \mathrm{\mid }\mathbf{a}\mathrm{\mid }+\mathrm{\mid }\mathbf{b}\mathrm{\mid }$.

The real numbers $x$, $y$ and $z$ satisfy $x+2y-2z=12$.

By choosing suitable vectors $\mathbf{a}$ and $\mathbf{b}$, find the minimum value of $x^2+y^2+z^2$ and the values of $x$, $y$ and $z$ for which it occurs.

Show that $(p+q+1{)}^2\le 3(p^2+q^2+1)$ for all $p,q\in \mathbb{R}$.