#SequencesSeries #Year12 #Advanced >[!info]- [Sequences and series | NSW Curriculum Website](https://curriculum.nsw.edu.au/learning-areas/mathematics/mathematics-advanced-11-12-2024/content/n12/fa8db67617) >- 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<br><br>- absolute value inequalities | - arithmetic sequences are linear functions<br>- working with limiting sums | | [[Exponential and Logarithmic Functions]] | - exponential functions | - geometric sequences are exponential functions | ## Sequences and series - 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 $a_1, a_2,\ldots, a_m$ that terminates at its mth term, for some whole number $m\geq1$, and an infinite sequence $a_1, a_2,\ldots$ that never terminates - Define the $n$th partial sum $S_n$ of a sequence $a_1, a_2,\ldots$ to be the sum of the first $n$ terms of the sequence: $S_n=a_1+a_2+\ldots+a_n$, for all whole numbers $n\geq1$ - Define a series formally as the sum of the terms of an infinite sequence and use the notation $a_1+a_2+a_3+\ldots$ for the series corresponding to the sequence $a_1,a_2,a_3\ldots$ - Use summation notation to represent the sum of terms $a_i$ to $a_j$ of a sequence where $j>i$, $\sum_{k=i}^{j}a_k=a_i+a_{i+1}+a_{i+2}+\ldots+a_{j-1}+a_j$ ## Arithmetic sequences and series - 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 $n\geq2$, 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 $n\geq1$, 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 $S_n=\frac{n}{2}[2a+(n-1)d]$ 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 ## Geometric sequences and series - 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 $n\geq2$, that is $a_n=ra_{n-1}$ for some non-zero real number r called the common ratio - Develop the formula $a_n=ar^{n-1}$ for the $n$th term of a GP, where $a$ is the first term and $n\geq1$, and use it to solve problems - Recognise that $a_n$ is an exponential function of $n$ in a GP - Prove by expansion $(x-1)(x^{n-1}+x^{n-2}+\ldots+x^2+x+1)=x^n-1$ for whole numbers $n\geq1$ - Develop the formula $S_n=\frac{a(1-r^n)}{1-r}$ for the sum of the first n terms of a GP where $r\neq1$, and use this formula to solve problems - Examine the behaviour of $a_n$ and $S_n$ as $n\to\infty$ for a GP when $|r|<1$