Sequences, series, and the arithmetic pattern

A sequence is an ordered list of terms, with positions usually labelled by positive integers. We write it as $\{u_n\}$: $u_n$ is the term in position $n$ and has the units of the quantity being modelled, while $n$ is just the term number, with no unit. A series is the sum of terms of a sequence.

An arithmetic sequence has a constant difference between each term and the one before it. That constant is the common difference, the fixed amount added as you move from one term to the next.

So the formula for the $n$th term is

Watch the $n-1$ carefully: to get from term $1$ to term $n$, you make $n-1$ jumps. In class I often use fence posts: $n$ posts create $n-1$ gaps.

If you plot the terms of an arithmetic sequence against $n$, the points lie on a straight line. The common difference is the gradient of that line when the horizontal step is one term.

Sums of arithmetic sequences

For an arithmetic sequence,

and, since $u_n=u_1+(n-1)d$, you can also write the same formula as

$S_n=\frac{n}{2}(u_1+u_n)$

The second form is often easier to use: number of terms multiplied by the average of the first and last terms. The two forms are equivalent, so choose the one that matches the information in the question.

A useful identity for any sequence, not only arithmetic ones, is

This is usually the quickest way to recover the term formula when a question gives you a formula for the sum.

Sigma notation

Sigma notation is a compact way to write a sum of terms generated by a rule. For example,

The capital sigma says, “add these values as $i$ runs from the lower limit to the upper limit”. The symbol is Greek because mathematical notation borrows from several alphabets; what matters is that it represents a sum, not a new operation to fear.

For an arithmetic sequence, you may see something like

$\sum_{i=1}^{n}\left(u_1+(i-1)d\right)$

That is just the arithmetic sequence term formula being added from $i=1$ to $i=n$.

Using technology without losing the mathematics

Spreadsheets, GDCs and graphing software are useful for generating terms, drawing the term graph, and checking a sum. In an examination, though, if you use technology, you still need to identify the first term and the common difference. The calculator can list values; it can’t explain the model for you.

A practical way to test whether data is arithmetic is to subtract consecutive values. Equal differences mean the sequence is arithmetic. If the differences are nearly equal, an arithmetic model may still be sensible, but you need to say that the common difference is approximate.

Applications and modelling

An arithmetic model is a mathematical model where a quantity changes by approximately the same amount in each equal time interval. Simple interest is a good example: the same amount of interest is added each year, so the balance increases arithmetically.

For simple interest,

The balance after $t$ years is $P+I$, so if $t$ increases by one each time, the increase per year is constant.

Real data is rarely perfectly arithmetic. When a table shows values rising by about the same amount, you may estimate an average common difference and use it for prediction. Be clear about interpretation: interpolation, predicting between known data values, is usually safer than extrapolation, predicting far beyond the data. A model that fits a few early values may later become unreasonable, such as a growth model that eventually predicts impossible negative values or unlimited growth.

The geometric pattern

A geometric sequence is a sequence where each term is found by multiplying the previous term by the same fixed number. That fixed number is the common ratio: the constant multiplier from one term to the next.

From this, the $n$th term is

$u_n=u_1r^{n-1}$

The exponent is $n-1$ because term $1$ has not been multiplied by $r$ yet. Term $2$ has one multiplier, term $3$ has two, and the pattern continues.

The common ratio can be positive, negative, greater than $1$, between $0$ and $1$, or less than $-1$. If the common ratio is negative, the signs alternate. Don’t discard a negative ratio just because a square root appeared in the working; if $r^2=9$, then $r=3$ or $r=-3$.

Sums of finite geometric sequences

A geometric series is the sum of the terms of a geometric sequence. For the first $n$ terms,

$S_n=\frac{u_1(r^n-1)}{r-1}$

or, equivalently,

$S_n=\frac{u_1(1-r^n)}{1-r},\quad r\ne1$

Both forms give the same value. I usually choose the first one when $r>1$, and the second one when $0<r<1$, because the arithmetic often comes out a little cleaner. If $r=1$, every term is $u_1$, so the sum is $nu_1$ rather than either of the fractions above.

Sigma notation for geometric sums

In sigma notation, the first $n$ terms of a geometric sequence can be written as

$\sum_{i=1}^{n}u_1r^{i-1}$

Read it like this: start with $i=1$, substitute into $u_1r^{i-1}$, then keep adding terms until $i=n$. It is the same geometric pattern, just in a more compact notation.

Generating and recognising geometric models

Spreadsheets, GDCs and graphing software can display geometric sequences as tables and graphs. In an examination, if you use technology, you still need to identify the first term and the common ratio. Use division, not subtraction: divide consecutive terms and check whether the ratio is constant or approximately constant.

Geometric sequences link naturally to exponential functions in Topic 2 and to regression in Topic 4. A geometric model is discrete, since it updates one term at a time. An exponential function is the continuous-looking graph that often represents the same multiplying behaviour.

Applications

Geometric models are used when a quantity is repeatedly multiplied by a fixed factor. Examples include population growth, salary increases and decreases, and the early spread of a disease when each stage produces a roughly constant multiplier. In physics, radioactive decay and capacitor charging or discharging also have this repeated-multiplier flavour.

In context, interpret the ratio carefully. A $5\%$ increase means multiplying by $1.05$, while a $5\%$ decrease means multiplying by $0.95$. The model can be useful for prediction, but only while the assumption of a constant multiplier remains reasonable.

From simple interest to compound interest

Compound interest is interest worked out on a balance that already includes earlier interest. So the pattern is geometric: at each compounding period, the balance is multiplied by the same factor.

If the nominal annual interest rate is $p\%$ and interest is compounded $k$ times per year, then

The value of $k$ changes with the compounding frequency: yearly gives $k=1$, half-yearly gives $k=2$, quarterly gives $k=4$, and monthly gives $k=12$. The exponent is $kt$ since that counts how many times interest is applied. That is why the compound interest formula uses an exponent based on the number of interest payments, rather than $n-1$ from the sequence formula.

In examinations you will not be asked to derive the compound interest formula, but you may need to use technology, including built-in financial packages. When you use a finance app or a GDC menu, keep track of the present value entry, the future value entry, and the number of compounding periods involved.

Annual depreciation

Depreciation is a decrease in value over time, usually modelled by multiplying by a factor less than $1$. If an item loses $p\%$ of its value each year, then

This is still a geometric model; the common ratio is just less than $1$.

If a value appreciates by $p\%$ annually, use $1+\frac{p}{100}$ instead of $1-\frac{p}{100}$. In comparison questions, for example when one asset depreciates while another appreciates, solve the inequality or equation carefully, then interpret whole years. If the model says equality occurs after $8.3$ years, then “at least as much after a complete number of years” usually means $9$ years.

Real value and inflation

Inflation is a sustained increase in general price levels, reducing the purchasing power of money. A nominal investment value may rise, but after allowing for inflation, its real value may rise more slowly or even fall.

A common calculation is

The calculation discounts the future amount back to present purchasing power.

Financial models aren't just button pressing. Loans, repayments, borrowing and lending all depend on assumptions about rates, time, risk and fairness. Different societies and institutions may view interest in different ways, so the mathematics is clean while the context may not be.

What it means for an infinite geometric series to converge

An infinite series is a sum with no final term. A convergent geometric series is an infinite geometric series whose partial sums get closer and closer to a finite value.

For a geometric series

$\sum_{i=1}^{\infty}u_1r^{i-1}$

the condition to check is

$|r|<1$

The modulus of a number is its distance from zero on the number line, so $|r|<1$ means $r$ lies between $-1$ and $1$. This condition matters more than the formula itself. If $|r|\ge 1$, the terms do not shrink to zero in the required way, so the infinite sum does not converge.

Sum to infinity

When $|r|<1$, the sum to infinity is

This comes straight from the finite geometric series formula in Section 1.3. In the finite formula, the part involving $r^n$ becomes negligible as $n$ becomes very large, but only when $|r|<1$. A shrinking multiplier gives a stable limiting sum.

Watch negative values of $r$ carefully. For example, $r=-\frac{1}{2}$ is perfectly acceptable because $\left|-\frac{1}{2}\right|<1$; the partial sums bounce above and below the limit but still settle down.

Interpreting infinity without pretending to list every term

Infinite geometric series let us reason about quantities we cannot literally finish adding. Repeating decimals are a neat example: a repeating block can be treated as a first term plus copies multiplied by powers of $\frac{1}{100}$, $\frac{1}{1000}$, or another suitable factor. Since that ratio has modulus less than $1$, the decimal can be written as a fraction.

One useful habit: whenever you use $S_\infty=\frac{u_1}{1-r}$, state or check $|r|<1$. The formula without the condition is like a seatbelt not clicked in; it looks ready, but it is not safe.