Extension Financial Mathematics – Series and sequence

This lesson comprises two (2) master classes focusing on:

• Arithmetic sequences
• Geometric sequences

Content:

MA-M1.2

• Know the difference between a sequence and a series
• Recognise and use the recursive definition of an arithmetic sequence: $T_n=T_{n−1}+d, T_1=a$
• Establish and use the formula for the $n^{th}$ term (where n is a positive integer) of an arithmetic sequence: $T_n=a+(n−1)d$, where $a$ is the first term and $d$ is the common difference, and recognise its linear nature
• Establish and use the formulae for the sum of the first n terms of an arithmetic sequence: $S_n=\frac{n}{2}(a+l)$ where $l$ is the last term in the sequence and $S_n=\frac{n}{2}\{2a+(n−1)d\}$
• Identify and use arithmetic sequence and arithmetic series in contexts involving discrete linear growth or decay such as simple interest

MA-M1.3

• Recognise and use the recursive definition of a geometric sequence: $T_n=rT_{n−1}, T_1=a$
• Establish and use the formula for the $n^{th}$ term of a geometric sequence: $T_n=ar^{n−1}$, where $a$ is the first term, $r$ is the common ratio and $n$ is a positive integer, and recognise its exponential nature
• Establish and use the formula for the sum of the first n terms of a geometric sequence: $S_n=\frac{a(1−r^n)}{1−r}=\frac{a(r^n−1)}{r−1}$
• Derive and use the formula for the limiting sum of a geometric series with $|r|<1: S=\frac{a}{1−r}$
  • understand the limiting behaviour as $n \to \infty$ and its application to a geometric series as a limiting sum
  • use the notation $lim_{n \to \infty}r^n=0$ for $|r|<1$