1.2 Arithmetic sequences and series questions for IB Maths AA

Find $u_{13}$.

Find the sum of the first $13$ terms.

Write down the common difference.

Find the number of seats in row 20.

The theatre has 20 rows. Find the total number of seats.

Consider the case $n=6$. Find $\mathrm{E}(X)$ and $\mathrm{V}\mathrm{a}\mathrm{r}(X)$.

Show that, for a spinner with $n$ sectors, $\mathrm{E}(X)={\displaystyle \frac{n+1}{2}}$.

Show that $\mathrm{V}\mathrm{a}\mathrm{r}(X)={\displaystyle \frac{n^2-1}{12}}$.

In a continuous model, the score is replaced by a random variable $Y$ with probability density function

Show that $\mathrm{E}(Y)={\displaystyle \frac{n+1}{2}}$ and $\mathrm{V}\mathrm{a}\mathrm{r}(Y)={\displaystyle \frac{n^2}{12}}$.

The value of $X$ can be obtained by rounding $Y$ to the nearest integer. Show that $\mathrm{V}\mathrm{a}\mathrm{r}(Y)-\mathrm{V}\mathrm{a}\mathrm{r}(X)$ does not depend on $n$, and state its value.

The spinner is replaced by a biased spinner with the same $n$ sectors. Its score $W$ has $\mathrm{P}(W=r)={\displaystyle \frac{2r}{n(n+1)}}$, for $r=1,2,\dots ,n$.

Show that $\mathrm{E}(W)={\displaystyle \frac{2n+1}{3}}$.

Show that $\mathrm{V}\mathrm{a}\mathrm{r}(W)={\displaystyle \frac{(n-1)(n+2)}{18}}$.

A continuous model for $W$ is the random variable $V$ with probability density function

Find $\mathrm{V}\mathrm{a}\mathrm{r}(V)$ in terms of $n$.

Find ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\frac{\mathrm{V}\mathrm{a}\mathrm{r}(W)}{\mathrm{V}\mathrm{a}\mathrm{r}(V)}}$, and explain briefly the significance of this result.

In a game, a player pays $$c$ to spin the biased spinner with $n=10$ sectors and then receives $$W$.

Find the value of $c$ for which the game is fair.

Find $u_{19}$.

Find the sum of the first $19$ terms.

Find $u_{10}$.

Find the sum of the first $10$ terms.

Write down the value of $R_1$, the value of $R_2$ and the value of $R_3$.

Suppose $k$ lines in general position are drawn, and then a $(k+1)$th line is added so that the $k+1$ lines are in general position. Explain why $R_{k+1}=R_k+k+1$.

Prove by mathematical induction that $R_n={\displaystyle \frac{n^2+n+2}{2}}$, for $n\in {\mathbb{Z}}^{+}$.

Now consider $n$ circles drawn in a plane so that every two circles meet at exactly two points and no three circles pass through the same point. Let $C_n$ be the number of regions into which the $n$ circles divide the plane. For example, $C_1=2$ and $C_2=4$.

Explain why $C_{k+1}=C_k+2k$, for $k\in {\mathbb{Z}}^{+}$.

Prove by mathematical induction that $C_n=n^2-n+2$, for $n\in {\mathbb{Z}}^{+}$.

Finally, $n$ points are placed on a circle and every pair of points is joined by a chord, with no three chords meeting at the same point inside the circle. Let $M_n$ be the number of regions into which the chords divide the inside of the circle. It is known that $M_1=1$, $M_2=2$, $M_3=4$, $M_4=8$ and $M_5=16$.

Suggest an expression for $M_n$ in terms of $n$.

It can be shown that $M_6=31$. Explain why this shows that your expression from part (e) is not correct.

It can be shown that $M_n={\displaystyle \left(\genfrac{}{}{0px}{}{n}{4}\right)}+{\displaystyle \left(\genfrac{}{}{0px}{}{n}{2}\right)}+1$, for $n\ge 4$.

Verify that this formula gives the correct values of $M_4$ and $M_6$.

By using the identity ${\displaystyle \left(\genfrac{}{}{0px}{}{m}{r}\right)}={\displaystyle \left(\genfrac{}{}{0px}{}{m-1}{r}\right)}+{\displaystyle \left(\genfrac{}{}{0px}{}{m-1}{r-1}\right)}$, show that, for $n\ge 5$,

Hence explain why $M_5=2^4$ and why $M_n<2^{n-1}$ for all $n\ge 6$.

Find the least value of $n$ for which $M_n>1000$.