Sequences and series: arithmetic, geometric and financial applications: study notes

1. Arithmetic sequences and series

An arithmetic sequence has a constant common difference $d = u_{n+1} - u_n$. The $n$th term is $u_n = u_1 + (n-1)d$ and the sum of the first $n$ terms is $S_n = \frac{n}{2}\left(2u_1 + (n-1)d\right) = \frac{n}{2}(u_1 + u_n)$. Both are in the booklet; what you must learn is how to use them. Two given terms give two linear equations in $u_1$ and $d$; subtracting them gives $d$ at once because $u_p - u_q = (p-q)d$. A term and a sum also give a pair of simultaneous equations. Note that $u_n = S_n - S_{n-1}$ for $n \geq 2$, and that for $d \neq 0$, $S_n$ is a quadratic in $n$ with no constant term (if $d = 0$ it is simply $S_n = nu_1$), so a condition like $S_n = 0$ or $S_n > 100$ becomes a quadratic equation or inequality in $n$. Simple interest is the standard application: a principal $P$ at $r\%$ per year grows by the fixed amount $\frac{Pr}{100}$ each year, so the values after each year form an arithmetic sequence with $u_1 = P + \frac{Pr}{100}$ and $d = \frac{Pr}{100}$. Be careful with the term index: the value after $n$ years is $P + \frac{nPr}{100}$, which is $u_n$ only if $u_1$ is the value after the first year.

2. Sigma notation

Sigma notation is not in the booklet. $\sum_{k=1}^{n} u_k = u_1 + u_2 + \dots + u_n = S_n$. The variable under the sign is a dummy index; the lower and upper limits give the first and last values of the index. Count the number of terms as (upper limit $-$ lower limit $+1$), so $\sum_{k=3}^{12}(2k+1)$ has $10$ terms, not $12$. To evaluate it, either write it as a difference $S_{12} - S_2$ of the full series, or treat it as its own arithmetic series with first term $7$, last term $25$ and $10$ terms, giving $\frac{10}{2}(7+25)=160$. To write a given series in sigma notation, identify $u_1$ and $d$ (or $r$), write the general term in terms of $k$ and fix the limits. For an infinite geometric series the upper limit is $\infty$: $\sum_{k=1}^{\infty} u_1 r^{k-1}$. Examiners accept any correct dummy variable, but the expression inside must produce exactly the listed terms.

3. Geometric sequences and series

A geometric sequence has a constant common ratio $r = \frac{u_{n+1}}{u_n}$, with $r \neq 0$. The $n$th term is $u_n = u_1 r^{n-1}$ and $S_n = \frac{u_1(r^n - 1)}{r-1} = \frac{u_1(1-r^n)}{1-r}$ for $r \neq 1$ (both in the booklet; the second form is convenient when $|r|<1$). Two terms give $r$ by division: $\frac{u_p}{u_q} = r^{p-q}$. If $p-q$ is even there are two possible values of $r$, positive and negative, and the question will usually tell you which to take or ask for both. When three consecutive terms are given as linear expressions in a parameter $x$, the arithmetic condition $u_2 - u_1 = u_3 - u_2$ gives a linear equation, while the geometric condition $u_2^2 = u_1 u_3$ gives a quadratic in $x$; reject any solution that makes a term zero. Applications: compound growth of money, a population growing by a fixed percentage each year ($r = 1 + \frac{p}{100}$), and depreciation ($r = 1 - \frac{p}{100}$). Finding $n$ from a target value needs logarithms: from $r^n > c$ take $\ln$ of both sides, remembering that if $0 < r < 1$ then $\ln r < 0$ and the inequality reverses. On a calculator paper, a table of values of $u_n$ or $S_n$ on the GDC is an accepted method provided you write down the two values either side of the target.

4. Infinite geometric series and convergence

If $|r| < 1$ then $r^n \to 0$ as $n \to \infty$, so $S_n \to \frac{u_1}{1-r}$. The booklet gives $S_\infty = \frac{u_1}{1-r}$ with the condition $|r|<1$, but you must be able to state the condition in words and use it as a reason: a geometric series with $u_1 \neq 0$ converges if and only if $-1 < r < 1$, endpoints excluded. The R mark in these questions is for writing this condition and applying it, for example when $r$ is an expression such as $\frac{x-2}{3}$, leading to the strict interval $-1 < x < 5$. A recurring decimal is a geometric series: $0.\dot{2}\dot{7} = 0.27 + 0.0027 + \dots$ with $u_1 = \frac{27}{100}$, $r = \frac{1}{100}$, so it equals $\frac{27/100}{99/100} = \frac{3}{11}$. When both $S_\infty$ and one term are known, eliminating $u_1$ gives a quadratic in $r$; check each root against $|r|<1$ before accepting it. Describe the behaviour of the partial sums if asked: for $0<r<1$ they approach the limit from one side (increasing if $u_1 > 0$, decreasing if $u_1 < 0$), for $-1<r<0$ they alternate above and below it.

5. Financial applications: compound interest and depreciation

The booklet formula is $FV = PV\left(1 + \frac{r}{100k}\right)^{kn}$, where $PV$ is the present value, $r$ the annual nominal rate as a percentage, $k$ the number of compounding periods per year ($k=1$ yearly, $2$ half-yearly, $4$ quarterly, $12$ monthly) and $n$ the number of years. Each period multiplies the balance by $1 + \frac{r}{100k}$, so the exponent counts periods, not years. Depreciation is not in the booklet: a value falling by $p\%$ per year is $V_n = V_0\left(1 - \frac{p}{100}\right)^n$, the same shape with a minus sign, so learn it. Typical tasks: find $FV$ to two decimal places; find the rate or the initial amount by rearranging; find the number of years (or months) for the value to exceed, or fall below, a target. For the last type set up the inequality, solve with logarithms or by the finance solver, then round up to the next whole period and state the answer in the units the question demands (complete years, months). An inflation adjustment compares two growth factors: the real growth factor is $\frac{1 + r/100}{1 + i/100}$, which is close to, but not equal to, $1 + \frac{r-i}{100}$.

6. Technology: GDC, finance solver and spreadsheets

On Paper 2 the finance solver on the GDC is an accepted method. Its inputs are the number of periods $N$, the annual rate $I\%$, the present value $PV$, the payment $PMT$ (zero for a lump sum), the future value $FV$, and the compounding settings. Sign convention matters: money paid in and money received have opposite signs, so a deposit of $5000$ with $PV = -5000$ gives a positive $FV$. Write the inputs out in full on the page: a visible, correct setup is what the method mark rewards, and if the final answer is wrong the listed inputs are the only evidence that the method was right. A sequence or table on the GDC (or a spreadsheet with a cell formula such as previous cell times $1.003$) is equally acceptable for 'how many years until' questions; show the two consecutive rows that bracket the target. Graphing software showing $y = S_x$ against $x$ helps to see where an arithmetic sum is maximised: the vertex of the parabola need not fall at an integer, and when it does not you must test the integers on each side rather than rounding. Paper 1 is a non-calculator paper, so technology is for revision only there: use it at home to check your exact Paper 1-style work, but in the exam the exact fraction or surd form, reached by hand, is what earns the marks.

Formulas

Worked examples

Can you...

• Find $d$ or $r$ from any two given terms by subtracting or dividing the term formulae, and solve the resulting pair of simultaneous equations?
• Convert a verbal condition ('first exceeds', 'is zero', 'falls below half') into an equation or inequality in $n$, solve it exactly or with logarithms, and justify the integer answer?
• Write a given series in sigma notation with correct limits and general term, and count the number of terms in a sum written in sigma notation?
• State the condition $|r|<1$ in words, apply it to a ratio containing a parameter to obtain a strict interval, and use $S_\infty$ for recurring decimals?
• Use the compound interest formula with $k$ periods per year and the depreciation formula, and record finance solver inputs or a GDC table so the method is visible?