HSC Maths Advanced Syllabus

The information explored in module 1 on arithmetic and geometric sequences, can be applied to the learning completion solving of series and finance problems. First a review of all important AP and GP features are important to discuss.

Finding the nth term of an AP

The formula to find the nth term of an arithmetic sequence is given by:

$$T_n=a+(n-1)d$$

Where $a$ is the first term in the sequence $(T_1)$ and $d$ is the common difference.

Can an AP be Formed Using 3 Numbers?

If the terms a, b and c are used in a sequence, as shown below:

$$a,b,c$$

We can use the following formulas to determine if the sequence is an AP, and if it is they can be used to find missing terms.

$$b-a = c-b\qquad \quad b=\frac{1}{2}(a+c)$$

Finding the nth Term of a GP

The formula to find the nth term of a geometric sequence is given by:

$$T_n = ar^{n-1}$$

Where $a$ is the first term in the sequence $T_1$ and $r$ is the common ratio.

Can a GP be Formed Using 3 Numbers?

If the terms e, f and g are used in a sequence, as shown below:

$$e,f,g$$

We can use the following formulas to determine if the sequence is an AP, and if it is they can be used to find missing terms.

$$\frac{f}{e}=\frac{g}{f}\qquad\qquad f^2=eg$$