An infinite geometric series has first term $18$ and sum to infinity $12$.

Consider the arithmetic series

where $m$ is an integer greater than $3$.

A geometric sequence has first term $u_1$ and common ratio $r$. It is known that

and

For a sequence $\{u_n\}$, the sum of the first $n$ terms is given by

where $p$ and $q$ are constants. It is known that $u_5=17$ and $S_8=100$.

The first three terms of an infinite geometric sequence are $x+6$, $x+2$ and $x+1$, respectively.

A deposit of $65536 is invested for one year at a nominal annual interest rate of $25\%$. Option A compounds the interest annually. Option B compounds the interest quarterly.

A quantity $Q$ is measured at equal time intervals. At times $t=0,1,2,3$, the values of $Q$ are $48,55,61,68$, respectively. An arithmetic model is to be used for prediction.

An infinite geometric series has first term $k$ and common ratio $k-2$, where $k$ is a real number.

An investment of 4200 euros earns interest at a nominal annual rate of $3.6\%$, compounded monthly. Inflation is modelled at $2.1\%$ per year.

For an arithmetic sequence, $S_5=80$ and $S_{12}=330$, where $S_n$ is the sum of the first $n$ terms.

A student deposits 150 dollars into a savings account at the end of each month. The account pays interest at a nominal annual rate of $4.8\%$, compounded monthly. The first deposit is made one month from now.

An infinite geometric series has first term $a$ and common ratio $r$, where $|r|<1$. The sum to infinity is $10$. The sum of the terms in even positions is $3$.

A charity receives donations according to the model

where $D_n$ is the donation, in dollars, received in week $n+1$, for $n=0,1,2,\ldots$.

An arithmetic sequence has first term $a$ and common difference $3$. The sum of the first $12$ terms is $330$.

Consider the arithmetic sequence defined by $u_k=2k+p$, where $p$ is a constant. It is given that the fourth term is $17$.

The distances, in kilometres, ridden by a cyclist on successive training days form a geometric sequence. The distance ridden on day $2$ is $24$ km and the distance ridden on day $5$ is $81$ km. The common ratio is positive.

An investment has initial value $4096 and increases by $25\%$ at the end of each year. A piece of equipment has initial value $15625 and depreciates by $20\%$ at the end of each year. Let $I_n$ and $E_n$ be their values after $n$ years.

An infinite geometric series has first term $6x-2$ and common ratio $\dfrac{x}{3}$, where $x\in\mathbb{R}$. Its sum to infinity is $6$.

A geometric sequence has first term $3$ and common ratio $r$, where $r>1$. Let $S_n$ denote the sum of the first $n$ terms. Given that $S_n=45$ and $S_{2n}=765$, determine $r$ and $n$.

A new outdoor theatre is built in rows. The first row has $22$ seats and each successive row has $4$ more seats than the previous row. For safety reasons, every fifth row has $6$ seats removed for access spaces.

A video is uploaded to a website. During the first hour it receives $1200$ views. During each subsequent hour, the number of views is modelled as $35\%$ greater than during the previous hour.

A pendulum moves through an arc length of $2.8$ metres on its first swing. Each subsequent swing has arc length $72\%$ of the previous swing.

The water level of a reservoir is recorded at the same time each week for five weeks. The data are not perfectly arithmetic, but an arithmetic model is to be used for prediction.

Two machines produce components each day. On day $n$, machine A produces

components and machine B produces

components.

The positive numbers $a$, $b$ and $12$ are consecutive terms of an arithmetic sequence. The positive numbers $a$, $b$ and $16$ are consecutive terms of a geometric sequence.

For a sequence $\{u_n\}$, the sum of the first $n$ terms is $S_n=An^2+Bn$, where $A$ and $B$ are constants. It is given that $u_3=11$ and $S_6=72$.

A geometric sequence has first term $2$ and common ratio $r$, where $r>1$. Let $S_n$ denote the sum of the first $n$ terms. It is known that $S_n=80$ and that the sum of the next $n$ terms is $6480$.

A savings account pays interest at $50\%$ per year, compounded annually. A fixed amount $D$ is deposited at the end of each year. Interest is credited immediately before each end-of-year deposit; therefore, each deposit first earns interest during the year following its deposit. Immediately after the fourth deposit is made, the account contains $2600.

A quantity $Q$ is recorded at equal time intervals $t=0,1,2,3,4$. The recorded values are $50,57,63,70,76$. An arithmetic model is to be formed using the mean of the consecutive differences.

A family starts a university fund. They deposit $2500$ dollars into an account at the end of each quarter. The account pays a nominal annual interest rate of $5.2\%$, compounded quarterly. The first deposit is made at the end of the first quarter.

Two graduates are offered different salary contracts. Contract A pays $42000$ dollars in the first year and increases by $1800$ dollars each year. Contract B pays $38000$ dollars in the first year and increases by $6\%$ each year.

A geometric sequence has first term $a$ and common ratio $r$, where $a>0$ and $r>1$. The sum of the first three terms is $21$. The sum of the fourth, fifth and sixth terms is $168$.

A retirement account initially contains $40000$ dollars. At the end of each year, the account earns interest at an annual rate of $4.5\%$, and then $2500$ dollars is withdrawn. Let $V_n$ be the value of the account immediately after the $n$th withdrawal.

An open-air theatre is built in horizontal rows. The number of seats in each row is modelled by an arithmetic sequence. A plan of the theatre shows that row $3$ contains $32$ seats and row $8$ contains $52$ seats. Let $u_n$ be the number of seats in row $n$, and let $S_n$ be the total number of seats in the first $n$ rows.

In an art installation, a beam of light reflects from a sequence of small mirrors. The intensity of the light patch after each reflection is modelled by a geometric sequence. The second light patch has intensity $18$ units and the fourth light patch has intensity $10.125$ units. Let $I_n$ be the intensity of the $n$th light patch.

A sculpture is built in layers. Let $S_n$ be the total number of blocks used after $n$ layers. An artist proposes the formula $S_n=3n^2+5n$ for $n\in\mathbb{Z}^+$. Let $u_n$ be the number of blocks used in layer $n$.