1.8 Infinite geometric series questions for IB Maths AA

Write down the probability that Kofi wins the game with his first toss.

Find the probability that Lena wins the game with her first toss.

Show that the probability that Kofi wins the game with his second toss is $\frac{4}{27}$.

Show that the probability that Kofi wins the game is $\frac{3}{5}$.

The rules are changed. Each player may now toss the coin at most three times. If none of the six tosses shows a head, the game is a draw.

Find the probability that the game is a draw.

Given that the game is not a draw, find the probability that Kofi wins.

Find the probability that Bruno wins with his first throw.

Show that the probability that Asha wins with her $n$th throw is $0.3\times {0.42}^{n-1}$.

Find the probability that Asha wins the game.

Given that the game ends within the first three throws, find the probability that Asha wins.

Find the least number of throws, $n$, such that the probability that the game lasts for more than $n$ throws is less than 0.01.

Write down the values of $F_3$, $F_4$, $F_5$ and $F_6$.

Let $f(x)={\displaystyle \frac{1}{1-x-x^2}}$, for $\mathrm{\mid }x\mathrm{\mid }<{\displaystyle \frac{\sqrt{5}-1}{2}}$.

Find $f(0)$ and $f^{\mathrm{\prime }}(0)$.

Show that $f^{\mathrm{\prime }\mathrm{\prime }}(0)=4$, and hence write down the first three terms of the Maclaurin series for $f(x)$.

Let $f(x)=a_0+a_{1}x+a_2{x}^2+a_3{x}^3+\cdots$.

By considering the identity $(1-x-x^2)(a_0+a_{1}x+a_2{x}^2+\cdots )=1$, show that $a_0=1$, $a_1=1$ and $a_n=a_{n-1}+a_{n-2}$ for $n\ge 2$.

Hence write down $a_n$ in terms of a Fibonacci number.

Let $\varphi ={\displaystyle \frac{1+\sqrt{5}}{2}}$ and $\psi ={\displaystyle \frac{1-\sqrt{5}}{2}}$.

Show that $1-x-x^2=(1-\varphi x)(1-\psi x)$.

Show that $f(x)={\displaystyle \frac{1}{\sqrt{5}}}({\displaystyle \frac{\varphi }{1-\varphi x}}-{\displaystyle \frac{\psi }{1-\psi x}})$.

By expanding each fraction in part (f) as a geometric series, show that $F_n={\displaystyle \frac{{\varphi }^n-{\psi }^n}{\sqrt{5}}}$ for $n\ge 1$.

State the condition on $x$ for both geometric series in part (g) to converge, and hence explain the domain given for $f$ in part (b).

Explain why $F_n$ is the integer nearest to ${\displaystyle \frac{{\varphi }^n}{\sqrt{5}}}$ for all $n\ge 1$.

Use your GDC and the result of part (i) to find $F_{30}$.

Find ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\frac{F_{n+1}}{F_n}}$.

Find the value of $k$.

Find the probability that $X$ is an even number.

Show that $r^4=0.44$, where $r$ is the common ratio.

Find the two possible values of the first term of the series.

A group consists of 4 boys and 5 girls. A team of 3 is chosen from the group.

By considering the number of boys in the team, show that

Explain, using a counting argument, why

where $m,n,r\in {\mathbb{Z}}^{+}$ and $r\le m$, $r\le n$.

By considering the coefficient of $x^r$ in $(1+x{)}^m(1+x{)}^n$, give a second justification of the result in part (b).

By choosing suitable values of $m$ and $r$ in part (b), show that ${\displaystyle \sum _{k=0}^n{\left(\genfrac{}{}{0px}{}{n}{k}\right)}^2=\left(\genfrac{}{}{0px}{}{2n}{n}\right)}$.

Verify the result in part (d) for $n=3$.

The numbers $\left(\genfrac{}{}{0px}{}{2n}{n}\right)$ are called central binomial coefficients. They also appear in the binomial expansion of $(1-4x{)}^{-\frac{1}{2}}$.

Show that the coefficient of $x^n$ in the expansion of $(1-4x{)}^{-\frac{1}{2}}$ is $\left(\genfrac{}{}{0px}{}{2n}{n}\right)$, for $n\in {\mathbb{Z}}^{+}$.

State the values of $x$ for which the expansion of $(1-4x{)}^{-\frac{1}{2}}$ is valid.

By considering $(1-4x{)}^{-\frac{1}{2}}\times (1-4x{)}^{-\frac{1}{2}}$, show that

Verify the result in part (g) for $n=2$.

A fair coin is tossed $2n$ times. Let $p_n$ be the probability of obtaining exactly $n$ heads, and let $p_0=1$.

Show that $p_n={\displaystyle \frac{1}{4^n}}{\displaystyle \left(\genfrac{}{}{0px}{}{2n}{n}\right)}$, and hence show that ${\displaystyle \sum _{k=0}^n{p}_k{p}_{n-k}=1}$.

Find the least value of $n$ for which $p_n<0.1$.