A-Level Maths 9709 - Pure Maths 1 - Series

1. Arithmetic Progressions (AP)

An arithmetic progression is a sequence of numbers where the difference between any two consecutive terms is constant. This constant difference is called the common difference, denoted by 'd'.

1.1 Formula for the nth term

The formula for the nth term (an) of an arithmetic progression is:

$a_n = a_1 + (n - 1)d$

where:

• $a_n$ is the nth term
• $a_1$ is the first term
• $n$ is the term number
• $d$ is the common difference

1.2 Sum of the first n terms

The sum (Sn) of the first n terms of an arithmetic progression is given by:

$S_n = \frac{n}{2}(a_1 + a_n)$

Alternatively, it can be expressed as:

$S_n = \frac{n}{2}(2a_1 + (n - 1)d)$

1.3 Example

Find the sum of the first 20 terms of the arithmetic progression 3, 8, 13, 18,...

Here, $a_1 = 3$, $d = 8 - 3 = 5$, and $n = 20$

$S_{20} = \frac{20}{2}(2(3) + (20 - 1)5)$

$S_{20} = 10(6 + 19 \times 5)$

$S_{20} = 10(6 + 95)$

$S_{20} = 10(101)$

$S_{20} = 1010$

2. Geometric Progressions (GP)

A geometric progression is a sequence of numbers where each term is multiplied by a constant value to obtain the next term. This constant value is called the common ratio, denoted by 'r'.

2.1 Formula for the nth term

The formula for the nth term (an) of a geometric progression is:

$a_n = a_1 \times r^{n - 1}$

where:

• $a_n$ is the nth term
• $a_1$ is the first term
• $n$ is the term number
• $r$ is the common ratio

2.2 Sum of the first n terms

The sum (Sn) of the first n terms of a geometric progression is given by:

$S_n = a_1 \times \frac{1 - r^n}{1 - r}$

This formula is valid when $r \neq 1$.

If $r = 1$, then $S_n = n \times a_1$

2.3 Example

Find the sum of the first 10 terms of the geometric progression 2, 6, 18, 54,...

Here, $a_1 = 2$, $r = \frac{6}{2} = 3$, and $n = 10$

$S_{10} = 2 \times \frac{1 - 3^{10}}{1 - 3}$

$S_{10} = 2 \times \frac{1 - 59049}{-2}$

$S_{10} = 2 \times \frac{-59048}{-2}$

$S_{10} = 2 \times 29524$

$S_{10} = 59048$

3. Binomial Expansion

The binomial expansion provides a formula for expanding expressions of the form $(a + b)^n$, where n is a non-negative integer.

3.1 General Term

The general term in the binomial expansion of $(a + b)^n$ is given by:

$T_{k+1} = \binom{n}{k} a^{n-k} b^k$

where:

• $T_{k+1}$ is the (k+1)th term
• $\binom{n}{k}$ is the binomial coefficient, calculated as $\frac{n!}{k!(n-k)!}$
• $a$ is the first term in the binomial expression
• $b$ is the second term in the binomial expression
• $n$ is the power to which the binomial expression is raised
• $k$ is the index of the term (starting from 0)

3.2 Example

Expand $(x + y)^3$ using the binomial expansion.

Here, $a = x$, $b = y$, and $n = 3$

$T_{k+1} = \binom{3}{k} x^{3-k} y^k$

1. For $k = 0$: $T_1 = \binom{3}{0} x^3 y^0 = 1 \times x^3 \times 1 = x^3$
2. For $k = 1$: $T_2 = \binom{3}{1} x^2 y^1 = 3 \times x^2 \times y = 3x^2y$
3. For $k = 2$: $T_3 = \binom{3}{2} x^1 y^2 = 3 \times x \times y^2 = 3xy^2$
4. For $k = 3$: $T_4 = \binom{3}{3} x^0 y^3 = 1 \times 1 \times y^3 = y^3$

Therefore, $(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3$