Geometric Sequences & Series (+sum of infinite sequences): notes and practice questions

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.

The first three terms of a geometric sequence are $u_{1} = 81, u_{2} = 54, \text{ and } u_{3} = 36$.

(a) Find the value of $r$, the common ratio of the sequence.

(b) Find the value of $n$ for which $u_n = 16$.

(c) Find the sum of the first 4 terms of the sequence.

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 large pendulum is set in motion. The arc length of its first swing, $l_1$, is 81 cm. The arc length of each subsequent swing is a constant fraction of the previous one, forming a geometric sequence $l_1, l_2, l_3, \dots$ .

The arc length of the second swing, $l_2$, is 54 cm.

(a) Find the common ratio of this sequence.

(b) Find the arc length of the pendulum's fifth swing.

(c) Calculate the total distance travelled by the pendulum in the first 4 swings.

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 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 $q$.

Consider a geometric sequence with first term 2 and common ratio 3.

$S_n$ is the sum of the first $n$ terms of the sequence.

(a) Find an expression for $S_n$, in the form $a^n+b$, where $a, b \in \mathbb{Z}$.

(b) Hence, show that $S_1 + S_2 + S_3 + \dots + S_n = \frac{3^{n+1}-2n-3}{2}$.

A set of 12 spherical Matryoshka dolls, $D_1, D_2, ..., D_{12}$, are designed to fit inside one another.

The smallest doll, $D_1$, has a radius of 2 cm.

The radius of each subsequent doll $D_{n+1}$ is 20% larger than the radius of doll $D_n$, for $n \in \mathbb{Z}^+, 1 \le n \le 11$.

(a) (i) Show that the volume of doll $D_n$ is $\frac{32\pi}{3} \left(\frac{216}{125}\right)^{n-1}$ cm$^3$.

(ii) Hence, find the mean volume of the twelve dolls, giving your answer in the form $p\pi \left(\left(\frac{216}{125}\right)^{12} - 1\right)$ cm$^3$, where $p \in \mathbb{Q}^+$.

(b) Find the median volume of the twelve dolls, giving your answer in the form $q\pi \left(\frac{216}{125}\right)^5$ cm$^3$, where $q \in \mathbb{Q}^+$.

Consider the arithmetic sequence $c, d, e, ...$, where $c, d, e \neq 0$.

Show that $c + e = 2d$.

Consider the geometric sequence $c, f, g, ...$, where $c, f, g \neq 0$.

Show that $f^2 = cg$.

The first term of both sequences is $c$. It is given that $e = g = 4$ and that $f$ is a real number.

Show that $d > 2$.

Consider the case where $c = 16$, $f > 0$ and $e = g = 4$.

(i) Write down the first four terms of the arithmetic sequence.

(ii) Write down the first four terms of the geometric sequence.

A new sequence $v_n$ is formed by combining the terms of the arithmetic sequence, $A_n$, and the geometric sequence, $G_n$, from part (d). The terms of $v_n$ are given by $v_n = A_n - \log_2(G_n)$.

(i) Show that $v_n$ is an arithmetic sequence and find its common difference.

(ii) Hence, find the value of $\sum_{n=1}^{12} v_n$.

All answers in this question should be given to four significant figures.

A popular online game offers players a 'Mystery Box' for £5. Each box contains a prize, with the probability distribution for the prize value $D$ shown in the following table. For example, the probability of a player receiving £$25$ is 0.04. The initial grand prize in the first week of the game is £$500$.

(a) Find the value of $c$.

(b) Determine whether purchasing a mystery box in the first week is a fair game. Justify your answer.

(c) If the grand prize is not won and continues to triple each week, while all other prize amounts and probabilities remain the same, write an expression in terms of $n$ for the value of the grand prize in the $n$th week of the game.

(d) The $w$th week is the first week in which a player is expected to make a profit from purchasing a mystery box. If a player purchases a mystery box in the $w$th week, their expected profit is $p$.

Find the value of $p$.

Consider the series $e^{2x} + k e^{2x} + \frac{1}{5}e^{2x} + ...$, where $x \in \mathbb{R}$ and $k \in \mathbb{R}, k \neq 0$.

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

(i) Show that $k = \pm \frac{1}{\sqrt{5}}$.

(ii) Given that $k > 0$ and the sum to infinity is $\frac{5+\sqrt{5}}{4}$, find the value of $x$.

(b) Now consider the case where the series is arithmetic.

(i) Show that $k = \frac{3}{5}$.

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

(iii) The sum of the first $n$ terms of the series is $-8e^{2x}$. Find the value of $n$.

The sum of the first $n$ terms of a geometric sequence is given by $S_n = \sum_{k=1}^{n} 3000 \left( \frac{3}{4} \right)^k$. This sequence models the amount of plastic (in kg) removed from a river by an environmental initiative each month.

(a) Find the amount of plastic removed in the first month, $u_1$.

(b) Calculate the total amount of plastic the initiative expects to remove if it continues indefinitely.

(c) The initiative aims to remove an amount of plastic that is within $0.05 \text{ kg}$ of its total expected removal goal. Find the least number of months, $n$, for which the total amount of plastic removed, $S_n$, is within $0.05 \text{ kg}$ of the total expected removal goal.

A start-up company's total revenue after $n$ months, in dollars, is given by the formula $S_n = 2n^2 + n$. The monthly revenues, $u_n$, form an arithmetic sequence.

(a) Calculate the total revenue after the first 5 months.

(b) Find the revenue generated in the 5th month.

(c) Find the revenue generated in the first month.

(d) Find an expression for the revenue in the $n$th month, $u_n$.

(e) A second company's monthly revenue, $v_n$, forms a geometric sequence. The revenue in the second month, $v_2$, is equal to the first company's revenue in its first month. The revenue in the fourth month, $v_4$, is equal to the first company's revenue in its seventh month.

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

(f) The second company experienced a period of rapid growth. Find the revenue of the second company in its 6th month.

A philanthropic foundation launches a donation matching program. For every dollar donated in the first month, they match a certain amount. In subsequent months, the matched amount decreases by a fixed proportion of the previous month's matched amount, forming a geometric sequence. The total matched amount over $n$ months is given by $S_n = \sum_{k=1}^{n} 6\left(\frac{3}{4}\right)^k$ dollars.

(a) Find the matched amount in the first month, $u_1$.

(b) Find the total matched amount if the program were to continue indefinitely, $S_\infty$.

(c) The foundation wants to ensure that the total matched amount is very close to its theoretical maximum. Find the least value of $n$ such that $S_\infty - S_n < 0.01$ dollars.

Consider a geometric sequence with $u_1 = 16$ and $u_4 = 2$.

(a) Find the common ratio, $r$.

(b) The following table shows the probability distribution of a discrete random variable $Y$ such that $P(Y=n) = \frac{u_n}{k}$, where $n \in \{1, 2, 3, 4\}$ and $k \in \mathbb{R}^+$.

Find the value of $k$.

The endowment fund of a university has an initial capital of $150\,000$. The fund invests this amount in a conservative bond fund that offers an interest rate of 6.5% per annum compounded annually.

(a) (i) Calculate the value of this investment after 8 years, giving your answer to the nearest hundred dollars.

(a) (ii) Determine the minimum number of years required for the investment to reach $300\,000$.

(b) The university considers investing in a more aggressive equity fund. Determine the minimum annual interest rate, $r\%$, compounded quarterly, required for the initial $150\,000$ investment to reach $300\,000$ after 7 years. Give your answer to two decimal places.

(c) A generous donor pledges to contribute to the endowment fund annually. The first contribution is $50\,000$, and each subsequent contribution is $\frac{3}{4}$ of the previous year's contribution.

(i) Show that the total amount contributed by the donor will never reach $250\,000$.

(ii) Find the amount the donor would need to contribute initially for the total contributions to reach $180\,000$ after 6 years. Give your answer to the nearest dollar.