Year 12 Maths Advanced (2027) Sequences and series

Sequences, Series & Sigma Notation

20 practice questions 0 video lessons Theory + worked examples

Learn sequences, series and sigma notation for NSW Year 12 Mathematics Advanced (NESA outcome MAV-12-03). A sequence is an ordered list of numbers, a series adds its terms, and sigma notation lets you write a long sum in one compact line.

You will learn to use a general term to find any term, tell finite lists from ones that go on forever, find a partial sum, and both read and write sums in sigma notation — the language used throughout the rest of the sequences and series work.

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Theory

A sequence is an ordered list of numbers and a series is their sum. This Year 12 Mathematics Advanced topic (MAV-12-03) covers the notation \(a_n\) for the \(n\)th term, the partial sum \(S_n\), and sigma notation \(\sum\).

The \(n\)th term is written \(a_n\) (position \(n=1,2,3,\dots\)). A sequence is finite if it stops, infinite if it continues forever.

The partial sum is \(S_n=a_1+a_2+\cdots+a_n\). Sigma notation \(\displaystyle\sum_{k=i}^{j}a_k\) adds the terms from \(k=i\) to \(k=j\) — that is \((j-i+1)\) terms. To recover one term from the totals, use \(a_n=S_n-S_{n-1}\).

The terms of \(\sum_{k=1}^{4}(2k+1)=3+5+7+9\).

\[S_n=a_1+a_2+\cdots+a_n\]

\[\sum_{k=i}^{j}a_k=a_i+a_{i+1}+\cdots+a_j,\qquad a_n=S_n-S_{n-1}\]

partial sum S n equals a1 plus a2 plus dots plus a n; a n equals S n minus S n minus 1

Method

Substitute \(n\) (or \(k\)) into the rule to list terms.
Add the listed terms for a series or a \(\sum\).
Use \(a_n=S_n-S_{n-1}\) to pull a single term from the partial sums.

Example 1 — Terms from a rule

Write the first three terms of \(a_n=3n-1\).

Solution

\(a_1\)	\(=\)	\(2\)
\(a_2\)	\(=\)	\(5\)
\(a_3\)	\(=\)	\(8\)

Example 2 — Evaluate a sum

Evaluate \(\displaystyle\sum_{k=1}^{4}(2k+1)\).

Solution

\(\ \)	\(=\)	\(3+5+7+9\)
	\(=\)	\(24\)

Example 3 — Term from \(S_n\)

If \(S_n=n^2+2n\), find \(a_3\).

Solution

\(a_3\)	\(=\)	\(S_3-S_2\)
	\(=\)	\(15-8=7\)

Example 4 — Sigma & sums

For \(a_n=n^2\): find \(a_4\), \(S_3\), and write \(2+4+6+8\) in sigma notation.

Solution

\(a_4\)	\(=\)	\(16\)
\(S_3\)	\(=\)	\(1+4+9=14\)
\(2+4+6+8\)	\(=\)	\(\displaystyle\sum_{k=1}^{4}2k\)

Common pitfalls

Count the terms. \(\sum_{k=2}^{6}\) has \(5\) terms, not \(4\).

The index need not start at \(1\). Read the bottom of the \(\sum\).

Sequence vs series. \(2,4,6\) is a sequence; \(2+4+6\) is a series.

Frequently asked questions

What is the difference between a sequence and a series?

A sequence is an ordered list of numbers, such as 2, 4, 6, 8. A series is what you get when you add the terms of a sequence, such as 2 plus 4 plus 6 plus 8.

What does sigma notation mean?

Sigma notation is a shorthand for a sum. The sum from k = i to j of a k means add the terms a i, a i plus 1, up to a j. The number of terms is j minus i plus 1.

How do you find a term from the partial sums?

Use a n equals S n minus S n minus 1, the new running total minus the previous one.

How many terms are in a sum written in sigma notation?

Count them as the top number minus the bottom number, plus one. For example the sum from k = 2 to 6 has 6 minus 2 plus 1, which is 5 terms.

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Arithmetic Sequences

Sequences, Series & Sigma Notation

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Theory