Arithmetic and geometric sequences and series, sigma notation, the sum to infinity of a convergent geometric series, and the binomial expansion for any rational index.

What this dot point is asking

WJEC wants you to work with arithmetic and geometric sequences and series, to use sigma notation, to find the sum to infinity of a convergent geometric series, and to expand $(1 + x)^n$ as a binomial series for any rational index $n$ (with its validity condition). The general binomial series extends the AS work, where $n$ had to be a positive integer.

The answer

Arithmetic sequences and series

An arithmetic sequence has a constant common difference $d$. The $n$th term is $a + (n-1)d$.

Geometric sequences and series

A geometric sequence has a constant common ratio $r$. The $n$th term is $ar^{n-1}$.

Sigma notation

The symbol $\sum$ denotes a sum: $\displaystyle\sum_{r=1}^{n} u_r = u_1 + u_2 + \cdots + u_n$. The lower and upper limits give the first and last index, and the expression after $\sum$ is the general term. A useful property is that $\sum$ is linear: $\displaystyle\sum (au_r + b) = a\sum u_r + bn$, so constant multiples factor out and a constant term added $n$ times contributes $bn$. For example $\displaystyle\sum_{r=1}^{4} (2r + 1) = 2(1+2+3+4) + 4(1) = 20 + 4 = 24$, which you can also check by listing the terms $3, 5, 7, 9$.

The binomial series for any index

For any rational $n$, the expansion is an infinite series valid for $|x| < 1$:

Examples in context

Example 1. A recurring decimal. The decimal $0.4444\ldots$ is the geometric series $0.4 + 0.04 + 0.004 + \cdots$ with $a = 0.4$ and $r = 0.1$. Its sum to infinity is $\dfrac{0.4}{1 - 0.1} = \dfrac{0.4}{0.9} = \dfrac{4}{9}$, confirming the fraction. The sum to infinity turns a recurring decimal into an exact fraction.

Example 2. Approximating a root. Using $(1 + x)^{1/2} \approx 1 + \tfrac{1}{2}x - \tfrac{1}{8}x^2$ with $x = 0.04$ gives $\sqrt{1.04} \approx 1 + 0.02 - 0.00002 = 1.0198$, very close to the true $1.0198$. The binomial series gives quick approximations to roots near $1$.

Try this

Q1. An arithmetic series has first term $5$ and common difference $3$. Find the sum of the first 10 terms. [3 marks]

• Cue. $S_{10} = \tfrac{10}{2}[2(5) + 9(3)] = 5(10 + 27) = 185$.

Q2. A geometric series has $a = 100$ and $r = 0.2$. Find the sum to infinity. [2 marks]

• Cue. $|r| < 1$, so $S_\infty = \dfrac{100}{1 - 0.2} = \dfrac{100}{0.8} = 125$.

Q3. Find the term in $x^2$ in the expansion of $(1 - x)^{-1}$. [2 marks]

• Cue. $(1 - x)^{-1} = 1 + x + x^2 + \cdots$, so the $x^2$ term is $x^2$ (coefficient $1$).

Q4. State the validity range for the expansion of $(1 + 3x)^{-1}$. [1 mark]

• Cue. Valid for $|3x| < 1$, that is $|x| < \tfrac{1}{3}$.