Sequences and Series

Sequences and series | NSW Curriculum Website

MAV-12-03 uses arithmetic and geometric sequences and series to model and solve problems

📖 Prior Knowledge

Content	Prior knowledge	Used for
Working with Functions	- linear functions - absolute value inequalities	- arithmetic sequences are linear functions - working with limiting sums
Exponential and Logarithmic Functions	- exponential functions	- geometric sequences are exponential functions

Define a sequence as an ordered list of objects
Use the notation $a_{n}$ , where $n$ is a positive integer, to represent the $n$ th term of a sequence
Distinguish between a finite sequence $a_{1}, a_{2}, \dots, a_{m}$ that terminates at its mth term, for some whole number $m \geq 1$ , and an infinite sequence $a_{1}, a_{2}, \dots$ that never terminates
Define the $n$ th partial sum $S_{n}$ of a sequence $a_{1}, a_{2}, \dots$ to be the sum of the first $n$ terms of the sequence: $S_{n} = a_{1} + a_{2} + \dots + a_{n}$ , for all whole numbers $n \geq 1$
Define a series formally as the sum of the terms of an infinite sequence and use the notation $a_{1} + a_{2} + a_{3} + \dots$ for the series corresponding to the sequence $a_{1}, a_{2}, a_{3} \dots$
Use summation notation to represent the sum of terms $a_{i}$ to $a_{j}$ of a sequence where $j > i$ , $k = i \sum j a_{k} = a_{i} + a_{i + 1} + a_{i + 2} + \dots + a_{j - 1} + a_{j}$

Define a sequence $a_{n}$ to be an arithmetic sequence, or arithmetic progression (AP), if every difference $a_{n} - a_{n - 1}$ of successive terms is the same where $n \geq 2$ , that is $a_{n} - a_{n - 1} = d$ for some constant $d$ called the common difference
Develop the formula $a_{n} = a + (n - 1) d$ for the $n$ th term of an AP, where a is the first term and $n \geq 1$ , and use it to solve problems
Recognise that $a_{n}$ is a linear function of $n$ in an AP
Develop the formula $S_{n} = \frac{n}{2} (a + a_{n})$ for the $n$ th partial sum of an AP, and use the formula to solve problems
Develop the formula $S_{n} = \frac{n}{2} [2 a + (n - 1) d]$ for the $n$ th partial sum of an AP, and use the formula to solve problems
Apply the formulas for arithmetic sequences and their partial sums to model and solve growth and decay problems involving a quantity that is a linear function of time

Define a sequence $a_{n}$ to be a geometric sequence, or geometric progression (GP), if every ratio $\frac{a _{n}}{a _{n - 1}}$ of successive terms is the same where $n \geq 2$ , that is $a_{n} = r a_{n - 1}$ for some non-zero real number r called the common ratio
Develop the formula $a_{n} = a r^{n - 1}$ for the $n$ th term of a GP, where $a$ is the first term and $n \geq 1$ , and use it to solve problems
Recognise that $a_{n}$ is an exponential function of $n$ in a GP
Prove by expansion $(x - 1) (x^{n - 1} + x^{n - 2} + \dots + x^{2} + x + 1) = x^{n} - 1$ for whole numbers $n \geq 1$
Develop the formula $S_{n} = \frac{a ( 1 - r ^{n} )}{1 - r}$ for the sum of the first n terms of a GP where $r \neq = 1$ , and use this formula to solve problems
Examine the behaviour of $a_{n}$ and $S_{n}$ as $n \to \infty$ for a GP when $∣ r ∣ < 1$