Advanced 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 \( |r|<1 \)