Edexcel IAL - Pure Maths 2- 4.2 Arithmetic Sequences and Series; Sum of First n Terms- Study notes  - New syllabus

Arithmetic Sequences and Series

An arithmetic sequence is a sequence in which the difference between consecutive terms is constant.

This constant difference is called the common difference, denoted by \( d \).

Arithmetic Sequence

An arithmetic sequence has the form:

\( a,\ a+d,\ a+2d,\ a+3d,\ \ldots \)

where:

• \( a \) is the first term
• \( d \) is the common difference 

Formula for the nth Term

The nth term of an arithmetic sequence is given by:

\( a_n = a + (n-1)d \)

This formula allows any term of the sequence to be found directly.

Arithmetic Series

An arithmetic series is the sum of the terms of an arithmetic sequence.

The sum of the first \( n \) terms is denoted by \( S_n \).

Sum of a Finite Arithmetic Series

The sum of the first \( n \) terms of an arithmetic sequence is given by:

\( S_n = \dfrac{n}{2}\,[2a + (n-1)d] \)

Alternatively, if the last term \( l \) is known:

\( S_n = \dfrac{n}{2}(a + l) \)

Proof of the Sum Formula

Consider the arithmetic series:

\( S_n = a + (a+d) + (a+2d) + \cdots + [a+(n-1)d] \)

Write the series in reverse order:

\( S_n = [a+(n-1)d] + [a+(n-2)d] + \cdots + a \)

Add the two expressions term by term:

\( 2S_n = n[2a + (n-1)d] \)

Divide both sides by 2:

\( S_n = \dfrac{n}{2}[2a + (n-1)d] \)

Hence proved.

Sum of the First n Natural Numbers

The sequence of natural numbers is:

\( 1,\ 2,\ 3,\ \ldots,\ n \)

This is an arithmetic sequence with:

• \( a = 1 \)
• \( d = 1 \)

Using the sum formula:

\( 1 + 2 + 3 + \cdots + n = \dfrac{n(n+1)}{2} \)

Sigma Notation \( \Sigma \)

Sigma notation is used to represent sums compactly.

The symbol \( \Sigma \) means “sum of”.

\( \sum_{k=1}^{n} a_k \)

This means the sum of the terms \( a_1 + a_2 + \cdots + a_n \).

Examples:

• \( \sum_{k=1}^{n} k = 1 + 2 + \cdots + n \)
• \( \sum_{k=1}^{n} (2k+1) \) represents the sum of odd numbers