Sequences and Series

Sequences and series was worth 8 marks on NESA's sample paper for the 2024 syllabus. It also underpins Financial mathematics, where annuity and loan formulas come straight from the geometric series sum, so it is worth getting fluent.

• Sequence and series language: $n$th term $a_n$, finite and infinite sequences, partial sum $S_n$, summation (sigma) notation.
• Arithmetic sequences (APs): common difference, $n$th term, two forms of the sum, modelling linear growth and decay.
• Geometric sequences (GPs): common ratio, $n$th term, the factorisation identity, the sum formula, limiting sums, modelling exponential growth and decay.

A reference sheet is provided in the HSC exam, but you must know these formulas and when each one applies.

A sequence is an ordered list $a_1,a_2,a_3,\dots$ and $a_n$ is the $n$th term ($n$ is a positive integer). The partial sum $S_n$ is the sum of the first $n$ terms. A series is the sum of the terms of a sequence.

Sigma notation writes a sum compactly. The number under the sigma is the first value of the counter and the number on top is the last:

Count terms carefully: ${\sum }_{k=i}^j$ has $j-i+1$ terms. This sum is an AP with 12 terms, first term 3 and last term 58, so it equals $\frac{12}{2}(3+58)=366$.

A useful link: for $n\ge 2$, $a_n=S_n-S_{n-1}$. This lets you find any term when you are given a formula for the sum.

An arithmetic sequence (AP) has a constant common difference $d=a_n-a_{n-1}$.

Here $a$ is the first term and $l=a_n$ is the last term. Use the first sum formula when you know the last term, the second when you know $d$.

Because $a_n=dn+(a-d)$, the terms of an AP are a linear function of $n$: plotted against $n$, they lie on a straight line with gradient $d$. That is why APs model quantities that change by the same amount each period, such as simple interest, straight-line depreciation or a savings plan that increases by a fixed amount.

Worked example 1

A theatre has 22 seats in the front row, and each row has 3 more seats than the row in front.

(a) The theatre has 18 rows. How many seats are in the back row, and in total?

(b) A new theatre uses the same pattern. What is the least number of rows that gives at least 1000 seats?

The positive root of $3n^2+41n-2000=0$ is

so $n$ must be a whole number of at least 20. Check: $S_{19}=931$ and $S_{20}=1010$. The answer is 20 rows. Showing the check at $n=19$ and $n=20$ earns the final mark in questions like this.

A geometric sequence (GP) has a constant non-zero common ratio $r=\frac{a_n}{a_{n-1}}$.

The terms of a GP are an exponential function of $n$. GPs model quantities that change by the same percentage each period: compound interest, population growth, declining-balance depreciation, or a drug washing out of the body.

The sum of a GP

The syllabus asks you to prove this identity by expanding the left-hand side (all the middle terms cancel):

Putting $x=r$ and multiplying by $a$ gives the sum of the first $n$ terms, for $r\ne 1$:

Both forms are equal. The second is tidier when $r>1$.

Limiting sums

If $-1<r<1$, then $r^n\to 0$ as $n\to \infty$, so the terms shrink to zero and $S_n$ approaches a finite value, the limiting sum:

If $|r|\ge 1$ there is no limiting sum. When a question has a variable ratio, such as $r=2x$, the condition for a limiting sum is an inequality to solve: $-1<2x<1$.

Recurring decimals are a classic use: $0.\dot{4}=0.4+0.04+\dots$ is a GP with $a=0.4$ and $r=0.1$, so it equals $\frac{0.4}{0.9}=\frac{4}{9}$.

Worked example 2

A ball is dropped from a height of 5 m. Each time it hits the ground it rebounds to 60% of the height it fell from.

(a) Find the total distance the ball travels before it comes to rest.

The first fall is 5 m. After that, each bounce goes up and comes down the same height: $3,1.8,1.08,\dots$ metres. This is a GP with $a=3$ and $r=0.6$, and $|r|<1$, so

(b) After which bounce is the rebound height first less than 10 cm?

The height after the $n$th bounce is $5(0.6{)}^n$ metres. Solve

The inequality sign flips because $\ln 0.6<0$. So it is the 8th bounce (heights: 14.0 cm after the 7th, 8.4 cm after the 8th).

Decide first whether the change is by a fixed amount (AP, linear) or a fixed percentage (GP, exponential). The syllabus just says growth and decay; these are typical contexts.

Here $r$ is the interest or depreciation rate per compounding period and $n$ is the number of periods. For 6% p.a. compounded monthly over 5 years, use $r=0.005$ and $n=60$.

A total over several years (total output, total amount saved) is a sum, so use $S_n$. A value in a particular year is a term, so use $a_n$. Mixing these up is the most common modelling error.

Worked example 3

A mine produces 52 000 tonnes of ore in its first year, and its output falls by 4% each year after that. Find its output in the 10th year and its total output over the first 10 years, to the nearest tonne.

The yearly outputs form a GP with $a=52000$ and $r=0.96$ (decay, so $r<1$).

Note the index is 9 (not 10) for the 10th term: the first year has no decrease yet. With $r<1$, the form $\frac{a(1-r^n)}{1-r}$ keeps everything positive.

• Using $r^n$ instead of $r^{n-1}$ in the $n$th term formula, or $n$ instead of $n-1$ in the AP formula.
• Using the limiting sum formula without checking $|r|<1$.
• Forgetting to flip the inequality when dividing by the log of a number less than 1.
• Giving a non-integer answer to "how many terms": round up and check the neighbouring whole numbers.
• Confusing a term (value in one year) with a sum (total over several years).
• In the bouncing ball problem, forgetting the first drop is counted once but every bounce counts twice.
• Using an annual rate with a monthly number of periods.

• Show that / prove: write the general term or the sum formula, substitute, and simplify to exactly what is asked. In the identity proof, expand fully and show the cancelling.
• State whether the sequence is an AP or a GP and give $a$ and $d$ (or $r$). Markers often award the first mark for this.
• For "least number of terms" questions, solve the equation, then test the integers either side. Logarithms are the expected method for GPs.
• Keep calculator values unrounded until the end, then round as the question asks (money to the nearest cent or dollar).
• The financial mathematics questions in MA12-FM are built on $S_n$ for a GP. Practise writing out the first few terms before jumping to a formula.

• MA12-FM Financial mathematics: annuities and reducing-balance loans are geometric series in disguise.
• MA11-EL Exponential and logarithmic functions: solving $r^n>k$ needs logarithms, and GPs are exponential functions of $n$.
• MA11-FN Working with functions: APs are linear functions; inequalities from limiting sum conditions use the same algebra.
• MA12-AC Applications of calculus: continuous exponential growth $Q=A{e}^{kt}$ is the calculus version of a GP.