A ball is dropped vertically from a height of $12$ m. After each impact with the ground, it rebounds to $65\%$ of the maximum height reached before that impact. Assume this pattern continues indefinitely.

A non-overlapping design is constructed in stages. At stage $1$, one square of area $64\text{ cm}^2$ is added. Each square added at one stage produces four new squares at the next stage, each having one ninth of its parent's area.

An automated machine performs repeated operating cycles. The first cycle lasts $40$ seconds, and each subsequent cycle lasts $92\%$ of the duration of the preceding cycle. Assume that the process continues indefinitely.

A runner compares two twelve-week training plans. Under plan A, she runs $5$ km in week $1$ and increases the weekly distance by $1.2$ km. Under plan B, she runs $5$ km in week $1$ and increases the weekly distance by $15\%$.

Consider the infinite series

For a real number $p$, consider the infinite geometric series

A geometric series has positive terms. Its sum to infinity is $50$, and the sum of its first four terms is $46.875$. Let its first term be $a$ and its common ratio be $r$.

A coastal restoration project plants seedlings in parallel strips. The first strip contains $125$ seedlings, and each subsequent strip contains $18$ more seedlings than the preceding strip.

A documentary receives $840$ views on the first day after its release. A model predicts that the daily number of views will increase by $16\%$ each day.

A stepped public artwork is completed in $18$ stages. The number of tiles added at stage $k$ is modelled by

An online magazine has $12\,500$ subscribers at the beginning of month $1$. The number of subscribers at the beginning of each subsequent month is modelled as $95.5\%$ of the number at the beginning of the preceding month.

A research team installs snow-collection trays along a mountain ridge. The first tray collects $18$ cm of snow during a season. The amount collected by each successive tray is modelled as $2.4$ cm more than the amount collected by the preceding tray. Let $u_n$ denote the modelled amount collected by tray $n$.

A circular installation contains concentric rings of lights. The innermost ring contains $30$ lights, and each successive ring contains $8$ more lights than the preceding ring.

The salinity of water in an experimental pond is measured at the end of each of six days. The measurements, in grams per litre, are

A researcher proposes an arithmetic model for the salinity.

Boxes are arranged in horizontal layers for a warehouse display. The bottom layer contains $96$ boxes. Each layer above contains $4$ fewer boxes than the layer immediately below it.

A pulsed lamp releases $18$ joules of energy in its first pulse. The energy released in each subsequent pulse is $78\%$ of the energy released in the preceding pulse.

A cooperative harvests $480$ tonnes of grain in year $1$. For the first six years, the annual harvest increases by $35$ tonnes each year. From year $7$ onwards, the harvest is modelled as increasing by $3\%$ each year, starting from the harvest in year $6$.

A digital signal processor records a signed response of $48$ units from the first reflection of a signal. Each subsequent reflection has signed response $-0.62$ times that of the preceding reflection. The reflections are assumed to continue indefinitely.

A pendulum travels $1.8$ metres during its first one-way swing. Each subsequent one-way swing has length $84\%$ of the preceding swing. Assume this pattern continues indefinitely.

A purification system initially contains $120$ milligrams of a contaminant. The first purification cycle removes $30$ milligrams. Each subsequent cycle removes $70\%$ of the amount removed in the preceding cycle.

A light installation is built in stages. At stage $1$, the added bulbs consume a total of $12$ watts. At each subsequent stage, five times as many bulbs are added, but each new bulb consumes one eighth of the power of a bulb added at the preceding stage.

A robot moves along a straight corridor towards a wall. Its first movement is $6$ metres. Each subsequent movement is $55\%$ of the preceding movement. The robot starts $14$ metres from the wall.

An operator digitizes historical documents during an $18$-hour project. In the first hour, $125$ documents are digitized. Owing to fatigue, the number digitized in each successive hour is modelled as $7$ fewer than in the preceding hour.

The numbers of tiles used in successive horizontal layers of a stepped wall form an arithmetic sequence. Layer $8$ contains $41$ tiles and layer $20$ contains $77$ tiles.

The lengths, in millimetres, of a seedling measured at the ends of six consecutive days are $82.0$, $85.1$, $87.7$, $91.0$, $93.8$ and $96.9$. A biologist proposes an arithmetic model for the measurements.

A marine restoration project attaches coral modules in stages. The area covered in stage $1$ is $45\ \text{m}^2$. The area covered in each successive stage is modelled as $18\%$ greater than in the preceding stage.

A photosensitive sensor receives a sequence of light pulses. The first pulse has exposure $900$ lux-seconds, and each subsequent pulse has $82\%$ of the exposure of the preceding pulse.

A filtration system initially contains $600$ litres of water. During each cycle, $13\%$ of the water present at the start of the cycle is removed, and the remainder passes to the next cycle.

A robot draws a path consisting of straight segments. The first segment has length $2.4$ m. Each subsequent segment has length $72\%$ of the preceding segment.

For a real number $x$, consider the infinite geometric series

A temperature-control algorithm starts with an estimate of $18^\circ\text{C}$. It then applies successive corrections. The first correction is $+5.4^\circ\text{C}$, and each subsequent correction is $-0.4$ times the preceding correction.

A drone surveys a straight boundary using a sequence of successively shorter flight legs. The first leg is $250$ metres. Each leg after the first is a constant fraction $q$ of the preceding leg, where $0<q<1$. The total distance flown during the first three legs is $610$ metres.

For integers $n\geq4$, consider the sum

An automatic control system makes successive signed adjustments to a temperature setting. The first adjustment is $20^\circ\text{C}$ and each later adjustment is $-0.4$ times the preceding adjustment. Let $S_n$ be the net adjustment after $n$ corrections.

A laser-cut pattern is made by removing regions from a sheet. At stage $1$, an area of $120\ \text{cm}^2$ is removed. At each later stage, the area removed is $\frac{3}{8}$ of the area removed at the preceding stage.

A signal processor records signed pulse areas. The first pulse area is $8$ volt-milliseconds, and each subsequent pulse area is $-0.55$ times the preceding pulse area.

Two online campaigns track the number of new subscribers gained each week. Campaign A gains $100$ subscribers in week $1$ and increases its weekly gain by $12\%$ each week. Campaign B gains $140$ subscribers in week $1$ and increases its weekly gain by $5\%$ each week.

A self-similar artwork is constructed in stages. The total area added in stage $1$ is $50\ \text{cm}^2$. From one stage to the next, the number of added pieces is multiplied by $3$, while the area of each piece is multiplied by $\frac14$. The total new boundary length in stage $1$ is $24$ cm; each new piece at the following stage has half the boundary length of its parent piece.

For a real parameter $p$, consider the infinite geometric series