Sum of infinite geometric sequences: notes and practice questions

A new type of eco-friendly battery is being tested for its long-term performance. In its first full charge cycle, the battery delivers a total capacity of 2500 mAh. Due to gradual degradation, the battery's maximum capacity for each subsequent charge cycle is only 92% of the capacity delivered in the previous cycle.

(a) Calculate the total cumulative charge capacity delivered by the battery over its first 8 full charge cycles.

(b) The manufacturer guarantees that the battery will deliver a total cumulative charge capacity of at least 31500 mAh over its entire lifespan. Determine if this battery meets the manufacturer's guarantee.

(c) Suggest a limitation of the given model for battery degradation.

A chemical spill has contaminated a section of a river. Environmental engineers are monitoring the concentration of a particular pollutant. Let $P(t)$, measured in milligrams per litre (mgL$^{-1}$), be the concentration of the pollutant, $t$ days after a new batch of pollutant is introduced. The rate at which the pollutant naturally degrades or is flushed away is modelled as directly proportional to its concentration, leading to the differential equation

$\frac{dP}{dt} = -kP$, where $k \in \mathbb{R}^+$.

The initial concentration is $P_0$ mgL$^{-1}$, $P_0 > 0$.

By solving the differential equation, show that $P = P_0 e^{-kt}$.

For the remainder of this question, you will consider this pollutant where it is known that $k = 0.15$. The first significant spill occurs at time $t = 0$ and it is assumed that before this there is no pollutant present in the river.

Find the time, in days, for this pollutant to reach 10% of its initial concentration.

The pollutant is added to the river every $T$ days due to regular discharges, and in constant amounts, such that the concentration of the pollutant is increased by an amount $P_0$ mgL$^{-1}$. To simplify the model, it is assumed that each time the pollutant is added, the concentration in the river increases instantaneously.

Show that the concentration of the pollutant is $P_0(1+e^{-0.15T} + e^{-0.30T})$ immediately after the third discharge is given.

Immediately after the $n^{th}$ discharge is given, the concentration of the pollutant is

$P_0(1+e^{-0.15T} + e^{-0.30T} + ... + e^{-0.15(n-1)T})$.

Show that this concentration can be expressed as $P_0\left(\frac{1-e^{-0.15nT}}{1-e^{-0.15T}}\right)$.

After the river has been subjected to these discharges for a long time, it is required to keep the pollutant concentration within a particular range to ensure ecological safety.

Let $H_n$ be the highest concentration of the pollutant in the river for the interval $(n-1)T < t < nT$.

Let $L_n$ be the lowest concentration of the pollutant in the river for the interval $(n-1)T < t < nT$.

This is shown in the following graph.

$H_\infty$ is defined as $\lim_{n\to\infty} H_n$ and $L_\infty$ is defined as $\lim_{n\to\infty} L_n$.

Find, in terms of $P_0$ and $T$, an expression for

$H_\infty$.

$L_\infty$.

Show that

$H_\infty - L_\infty = P_0$.

$\frac{1}{0.15} \ln\left(\frac{H_\infty}{L_\infty}\right) = T$.

It is known that this pollutant is considered safe if the long-term concentration never exceeds $0.50 \text{ mgL}^{-1}$ and ecologically effective if it never drops below $0.10 \text{ mgL}^{-1}$.

Hence, for this pollutant, find a suitable value for

$P_0$.

$T$.

For the values of $P_0$ and $T$ found in part (g), find the proportion of time for which the concentration of the pollutant is at least $0.10 \text{ mgL}^{-1}$ between the first and second discharges.

Suggest a reason why the environmental regulations might specify a different value for $T$ to that found in part (g)(ii).

A specialized bouncy ball is dropped from a certain height. After the first bounce, it reaches a maximum height of $u_1$ meters. After each subsequent bounce, the maximum height it reaches is a constant fraction of the previous maximum height.

The first bounce reaches a height of 3 meters. The sum of all the maximum heights reached after each bounce (i.e., $u_1 + u_2 + u_3 + \dots$) is 15 meters.

Find the common ratio, $r$, for the sequence of maximum bounce heights.

Find the least value of $n$ such that the maximum height reached after the $n$-th bounce, $u_n$, is less than 0.1 meters.

A specialized bouncy ball is dropped from a height of 16 meters. After the first bounce, it reaches a height of 8 meters. The height reached after each subsequent bounce is half of the height reached after the previous bounce.

(a) Calculate the height the ball reaches after its 4th bounce.

(b) Find the total vertical distance the ball has traveled upwards after 6 bounces.

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

(a) A newly developed computational model processes information in a series of iterative steps. The amount of new information processed in each step forms a geometric sequence. If the total amount of information processed by the model over an infinite number of steps is three times the amount processed in the first step, calculate the common ratio of this sequence.

(b) In a different scenario for the same computational model, the total amount of information processed over an infinite number of steps is $\frac{2}{3}$ times the amount processed in the first step. Calculate the common ratio in this case.

(c) Determine whether it is possible for the total amount of information processed by the model over an infinite number of steps to be $\frac{1}{3}$ times the amount processed in the first step. Justify your answer.

A group of archaeologists is excavating an ancient site and finds a collection of eight inscribed clay tablets. Five of these tablets contain mythological tales (M) and three are historical records (H). Dr. Anya Sharma and Dr. Ben Carter decide to play a game: they take turns drawing a tablet from a bag, without replacement. The first person to draw a historical record tablet wins.

(a) Dr. Anya Sharma plays first. Find the probability that she wins.

The game is now changed so that the tablet chosen is replaced after each turn. Dr. Anya Sharma still plays first.

(b) Determine whether the probability of Dr. Anya Sharma winning has changed.

A signal generator produces a sequence of complex voltage signals, $V_n$, where $n$ is the signal number. The first signal is $V_1 = 10$ volts. Each subsequent signal is generated by multiplying the previous signal by a complex factor $k = \frac{1}{2} + \frac{1}{2}i$.

Write down the value of $V_2$.

Write down the value of $V_3$.

The engineering team claims that the sequence of signal magnitudes $|V_1|, |V_2|, |V_3|, \dots$ also forms a geometric sequence.

Show that this claim is correct, stating the exact value of the common ratio for this sequence.

Hence, find the sum of the infinite sequence of magnitudes $|V_1|, |V_2|, |V_3|, \dots$.

An infinite geometric series has first term $u_{1}=k+1$ and second term $u_{2}=k^2-1$, where $k$ is a real number and $k>1$.

(a) Find the common ratio, $r$, in terms of $k$.

(b) Find the range of values of $k$ for which the sum to infinity of the series exists.

(c) Find the value of $k$ when $S_{\infty}=18$.