Year 12 Maths Advanced (2027) Sequences and series

Sequences, Series & Sigma Notation

20 practice questions 0 video lessons Theory + worked examples

NSW · Year 12 Mathematics Advanced · Sequences and series

Sequences, series and sigma notation introduce ordered lists of numbers and their sums, using \(T_n\) for the \(n\)th term and \(\displaystyle\sum\) notation to write a series compactly.

Part of the NSW Year 12 Mathematics Advanced course, in the Sequences and series focus area of the 2024 syllabus. Work through practice questions with fully worked solutions and video lessons, or scroll down for the theory summary and worked examples.

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Practice questions

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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