HSC Maths Advanced Syllabus

How to sum a Sequence

Adding the terms of a sequence is known as finding the sum of a series. This is denoted by the term $S_n$, where:

$$S_n=T_1+T_2+T_3+...+ T_n$$

In this case $n$ **represents the number of terms you are adding in the sequence. For example,

$$S_2=T_1+T_2$$

Sigma notation

Sigma notation is just an alternative way of displaying a sequence:

$$\sum_{n=3}^{9}T_n=T_3+T_4+...+T_9$$

To read this correctly, we must know that the number on-top of the greek letter sigma ($\sum$) is the term of the sequence the calculation is finishing at, while the term at the bottom of ($\sum$) is the term at which the calculation begins.

Summing an AP

If you are finding the summation of an arithmetic series, where each term is obtained by adding a constant difference $d$ to the previous term, the sum of the first n terms ($S_n$) is given by:

$$S_n = \frac{n}{2}(a+l)\qquad or \qquad S_n=\frac{n}{2}(2a+(n-1)d)$$

where $a$ is the first term of the sequence, $l$ is the last term, and $d$ is the common difference. Both formulas can be used depending on the scenario.

Summing a GP

For a geometric sequence, where each term is found by multiplying the previous term by a constant ratio, the sum of the first n terms is found using:

$$S_n=\frac{a(r^n-1)}{r-1}\qquad(when\quad r>1)\\or\\S_n=\frac{a(1-r^n)}{1-r}\qquad(when\quad r<1)$$

where $a$ is the first term of the sequence and $r$ is the common ratio.