Arithmetic Sequences & Series (+Sigma notation): notes and practice questions

(a) A financial analyst observes the annual profit (in millions of dollars) of a startup company. The profits for the first few years form an arithmetic sequence. The profit in the 2nd year, the 5th year, and the 11th year are observed to form a geometric sequence.

Given that the profit in the 1st year was 2 million dollars and the common difference of the arithmetic sequence is non-zero, find the common difference, $d$, of the arithmetic sequence.

(b) Find the common ratio, $r$, of the geometric sequence.

An outdoor amphitheater is being designed. The first row of seating has 25 seats. Each subsequent row has 3 more seats than the row in front of it. The last row has 91 seats.

(a) Find the number of rows of seats in the amphitheater.

(b) Find the total seating capacity of the amphitheater.

(a) A research team, 'Team Alpha', is monitoring a specific environmental parameter. On the first day, they collected 50 GB of data. Due to increasing efficiency, they manage to collect an additional 8 GB of data each subsequent day.

Calculate the total amount of data collected by Team Alpha during the first 15 days of their research.

(b) Another research team, 'Team Beta', is monitoring a different environmental parameter. On their first day, they collected 40 GB of data. Due to the nature of their data collection method, the amount of data they collect increases by 5% each subsequent day.

Calculate the total amount of data collected by Team Beta during the first 15 days of their research.

Consider the series $\sin\theta + k\sin\theta + \frac{1}{2}\sin\theta + \dots$, where $0 < \theta < \frac{\pi}{2}$ and $k \in \mathbb{R}, k \neq 0$.

Consider the case where the series is geometric.

(a) (i) Show that $k = \pm \frac{1}{\sqrt{2}}$.

(a) (ii) Given that $k > 0$ and the sum to infinity is $\sqrt{2}$, find the value of $\sin\theta$.

Now consider the case where the series is arithmetic with common difference $d$.

(b) (i) Show that $k = \frac{3}{4}$.

(b) (ii) Write down $d$ in the form $c\sin\theta$, where $c \in \mathbb{Q}$.

(b) (iii) The sum of the first $n$ terms of the series is $-14\sin\theta$.

Find the value of $n$.

A specialized superball is dropped from a height of 5 meters. After each bounce, it reaches a height that is a constant fraction of the previous height. After the third bounce, the ball reaches a height of 1.08 meters.

(a) Calculate the common ratio of the heights reached by the ball after each bounce.

(b) Determine the total vertical distance the ball travels downwards from the moment it is dropped until it theoretically comes to rest.

Consider the series $e^{2x} + p e^{2x} + \frac{1}{9}e^{2x} + ...$, where $x \in \mathbb{R}$ and $p \in \mathbb{R}, p \ne 0$.

(a) Consider the case where the series is geometric.

(i) Show that $p = \pm \frac{1}{3}$.

(ii) Hence or otherwise, show that the series is convergent.

(iii) Given that $p < 0$ and $S_\infty = \frac{3}{4}e^2$, find the value of $x$.

(b) Now consider the case where the series is arithmetic with common difference $d$.

(i) Show that $p = \frac{5}{9}$.

(ii) Write down $d$ in the form $k e^{2x}$, where $k \in \mathbb{Q}$.

(iii) The sum of the first $n$ terms of the series is $-10e^{2x}$.

Find the value of $n$.

A new company makes a profit of $50000 in its first month of operation. The profit is found to decrease by $1250 each subsequent month, forming an arithmetic sequence.

(a) Find the first month in which the company makes a loss (a negative profit).

(b) Calculate the maximum total profit the company could make.

By using an appropriate substitution, show that $\int \sin(\sqrt{x}) \, dx = 2\sin(\sqrt{x}) - 2\sqrt{x} \cos(\sqrt{x}) + C$.

The following diagram shows part of the curve $y = \sin(\sqrt{x})$ for $x \ge 0$.

The curve intersects the x-axis at $x_1, x_2, x_3, \dots$.

The nth x-intercept of the curve, $x_n$, is given by $x_n = n^2 \pi^2$, where $n \in \mathbb{Z}^+$.

Write down an expression for $x_{n+1}$.

The regions bounded by the curve and the x-axis are denoted by $R_1, R_2, R_3, \dots$ as shown on the diagram.

Calculate the area of region $R_n$.

Give your answer in the form $(an+b)\pi$, where $a, b \in \mathbb{Z}^+$.

Hence, show that the areas of the regions $R_1, R_2, R_3, \dots$ form an arithmetic sequence.

A function $f$ is defined by $f(x) = x^3 - 6x^2 + cx + d$, where $c, d \in \mathbb{R}$. The equation $f(x)=0$ has three real roots, $\alpha, \beta$ and $\gamma$.

(a) Write down the value of $\alpha + \beta + \gamma$.

A polynomial $P$ is defined by $P(z) = z^5 - 8z^4 + pz^3 + qz^2 + rz - 12$, where $p, q, r \in \mathbb{R}$. The roots of $f(x)=0$ are also roots of the equation $P(z)=0$. It is given that $z=1-i$ is a root of $P(z)=0$.

(b) Find the other non-real root of $P(z)=0$, giving a reason for your answer.

(c) Find the value of $\alpha\beta\gamma$.

(d) It is given that the roots $\alpha, \beta, \gamma$ form an arithmetic progression. Find the values of $\alpha, \beta$ and $\gamma$.

A company's profit in the $n$-th month after it was founded is given by the $n$-th term, $u_n$, of an arithmetic sequence. The total profit after $n$ months is given by $S_n = 10n - n^2$, in thousands of dollars.

(a) (i) Find the total profit after the first 3 months.

(a) (ii) Find the profit made in the 3rd month.

(b) Find the profit made in the first month.

(c) Find an expression for the profit in the $n$-th month, $u_n$, in terms of $n$.

(d) A second company's profit in the $n$-th month is given by the $n$-th term, $v_n$, of a geometric sequence. The profit in the second month, $v_2$, is equal to the first company's profit in the first month, $u_1$. The profit in the fourth month, $v_4$, is equal to the first company's profit in the fifth month, $u_5$.

Find the possible values of the common ratio, $r$.

(e) Given that the second company's profit is always positive, find the profit made in its third month, $v_3$.

Consider the function $f(x) = a^x$ where $x, a \in \mathbb{R}$ and $a > 1$. The graph of $f$ contains the point $(\frac{3}{2}, 27)$.

(a) Show that $a = 9$.

(b) Write down an expression for $f^{-1}(x)$.

(c) Find the value of $f^{-1}(\frac{1}{81})$.

Consider the arithmetic sequence $\log_9 8, \log_9 p, \log_9 q, \log_9 27$, where $p > 1$ and $q > 1$.

(i) Show that $8, p, q$ and $27$ are four consecutive terms in a geometric sequence.

Consider the arithmetic sequence $\log_9 8, \log_9 p, \log_9 q, \log_9 27$, where $p > 1$ and $q > 1$.

(ii) Find the value of $p$ and the value of $q$.

A graphic designer creates a sequence of patterns. The total number of dots in the first $n$ patterns is given by the formula $T_n = an^2 + bn$, where $a$ and $b$ are constants.

It is known that the total number of dots in the first 3 patterns is 33, and the total number of dots in the first 5 patterns is 75.

(a) Find the value of $a$ and the value of $b$.

(b) Find the number of dots in the fifth pattern.

A new theatre is being designed. The number of seats in each row forms an arithmetic sequence. The total number of seats in the first $n$ rows, denoted by $S_n$, is given by the formula $S_n = an^2 + bn$, where $a$ and $b$ are constants.

It is known that the total number of seats in the first 4 rows is 70, and the total number of seats in the first 6 rows is 135.

(a) Find the value of $a$ and the value of $b$.

(b) Find the number of seats in the 6th row.

A chemical engineer is studying the concentration of a certain byproduct in a reaction mixture over time. The concentration is modelled by the function $C_1(t) = t e^{-2t}$, where $t \ge 0$ represents time in hours.

(a) Sketch the graph of $y = C_1(t)$, stating the coordinates of the maximum concentration.

(b) The total amount of byproduct accumulated in the first $b$ hours is given by the integral $\int_0^b C_1(t) dt$.

Show that $\int_0^b t e^{-2t} dt = \frac{1}{4} (1 - (2b+1)e^{-2b})$.

(c.i) The total amount of byproduct accumulated indefinitely, $A_1$, is given by $\lim_{b \to \infty} \int_0^b C_1(t) dt$.

Use l'Hôpital's rule to find $\lim_{b \to \infty} \frac{1 - (2b+1)e^{-2b}}{4}$. You may assume that the condition for applying l'Hôpital's rule has been met.

(c.ii) Hence write down the value of $A_1$.

(d) The total amount of byproduct accumulated indefinitely for a general function $C_n(t) = t^n e^{-2t}$ is denoted by $A_n = \int_0^\infty t^n e^{-2t} dt$.

You are given that $A_2 = 0.25$ and $A_3 = 0.375$.

(i) Use your graphic display calculator, and an appropriate value for the upper limit, to determine the value of $A_4$.

(ii) Determine the value of $A_5$.

(e) Suggest an expression for $A_n$ in terms of $n$, where $n \in \mathbb{Z}^+$.

(f) Use mathematical induction to prove your conjecture from part (e). You may assume that, for any value of $m$, $\lim_{t \to \infty} t^m e^{-2t} = 0$.

This question asks you to examine linear and quadratic functions constructed in systematic ways using geometric sequences.

Consider the function $L(x) = mx + c$ for $x \in \mathbb{R}$ where $m, c \in \mathbb{R}$ and $m, c \neq 0$.

Let $r \in \mathbb{R}$ be the root of $L(x) = 0$.

If $m, r$ and $c$, in that order, are in geometric sequence, then $L(x)$ is said to be a GS-linear function.

Show that $L(x) = 2x + 8$ is a GS-linear function.

Consider $L(x) = mx + c$.

Show that $r = -\frac{c}{m}$.

Given that $L(x)$ is a GS-linear function, show that $L(x) = mx + m^3$.

State any further restrictions on the value of $m$, assuming the common ratio $q \neq \pm 1$.

There are only two integer values of $m$ (excluding $m=0$) for which $L(x) = mx+m^3$ is a GS-linear function with a common ratio $q \neq \pm 1$. One of these gives $L(x) = 2x+8$.

Use part (b) to determine the other GS-linear function with integer values of $m, r$ and $c$ that satisfies the condition.

Consider the function $Q(x) = ax^2 + bx + c$ for $x \in \mathbb{R}$ where $a \in \mathbb{R}, a \neq 0$ and $b, c \in \mathbb{R}$.

Let $r_1, r_2 \in \mathbb{R}$ be the roots of $Q(x) = 0$.

Write down an expression for

(i) the sum of roots, $r_1 + r_2$, in terms of $a$ and $b$.

(ii) the product of roots, $r_1 r_2$, in terms of $a$ and $c$.

If $a, b, c$ are in arithmetic sequence, AND $r_1, r_2$ are in geometric sequence, then $Q(x)$ is said to be a GS-Quadratic function.

Given that $Q(x)$ is a GS-Quadratic function, show that $qr_1^2 + 2(1+q)r_1 + 1 = 0$, where $q = r_2/r_1$.

Hence or otherwise, show that $q=1$ or $q=-1$.

Consider the case where $q=1$.

Determine the two GS-Quadratic functions that satisfy this condition, given that $a=1$. Give your answers in the form $x^2 + Bx + C$.

Consider the case where $q=-1$.

Determine the two GS-Quadratic functions that satisfy this condition, given that $a=\pm 2$.