Arithmetic Sequences & Series (Edexcel IGCSE Further Maths)

Arithmetic Sequences

Spec point — 5B

What is an arithmetic sequence?

• In an arithmetic sequence, the difference between consecutive terms in the sequence is constant

  • This means a common difference is added to each term to get the next term

• The first term of the sequence is denoted by $a$

• The common difference is denoted by $d$

  • For example, 1, 4, 7, 10, … is an arithmetic sequence with the rule ‘start at one and add three to each number’

    • The first term,$a$, **is 1
    • The** common difference, $d$, is 3

• An arithmetic sequence can be

  • increasing(positive common difference), or
  • decreasing(negative common difference)

• Terms in an arithmetic sequence can be referred to

  • by the letter $u$ with

  • a subscript corresponding to its place in the sequence

    • e.g. $u_{1}=a$ is the first term, $u_{9}$ is the ninth term, $u_{n}$ is the $n$^th term, etc.

Arithmetic Series

Spec point — 5B

What is an arithmetic series?

• When the terms of an arithmetic sequence are added together, that is known as an arithmetic series

  • The terms (1st term, 2nd term, 3rd term, etc.) are exactly the same in the sequence and series
  • But with series we're most interested in what happens when the terms are added together

How do I find a term in an arithmetic series?

• The n^thterm formula for an arithmetic sequence is

$u_{n}=a+(n-1)d$

• Where $a$* *is the first term, and $d$ is the common difference

• This is not given on the exam formula sheet, so make sure you know it

• The formula allows you to find any term in the arithmetic series

  • Enter the values of $a$, $r$ and $n$ and calculate the value of $u_{n}$

• Sometimes you will be given a term ($u_{n}$) and asked to find$a$or$d$

  • Substitute the information you have into the formula and solve the equation

• Sometimes you will be given two terms and asked to find both$a$and$d$

  • Substitute the information into the formula and set up a pair of simultaneous equations

    • Then solve the simultaneous equations

How do I find the sum of an arithmetic series?

• An arithmetic series is the sum of the terms in an arithmetic sequence

  • For the arithmetic sequence 1, 4, 7, 10, … the arithmetic series is 1 + 4 + 7 + 10 + …

• Use the following formula to find the sum of the first $n$ terms of an arithmetic series:

$S_{n} = \frac{n}{2} [ 2 a + ( n - 1 ) d ]$* *

• $a$_{* *}is the first term

• $d$ is the common difference

• The formula is given on the exam formula sheet

  • So you don't need to remember it
  • But you do need to know how to use it!

• A question will often give you the sum of a certain number of terms and ask you to find the value of$a$or$d$

  • Substitute the information you have into the formula and solve the equation

Exam Hint

• The formula for the sum of an arithmetic series is on the exam formula sheet

  • But the n^th term formula is not on the formula sheet

• Simultaneous equations are often needed within arithmetic series questions

  • Make sure you are confident solving them!

Worked Example The fourth term of an arithmetic series is 10 and the ninth term is 25. Find the first term and the common difference of the series.

Put the information for the fourth and ninth terms into the n^th term formula $u_{n} = a + ( n - 1 ) d$

For $u_{4}$

$\begin{gathered} a + ( 4 - 1 ) d & = 10 \\ a + 3 d & = 10 \end{gathered}$

For $u_{9}$

$\begin{gathered} a + ( 9 - 1 ) d & = 25 \\ a + 8 d & = 25 \end{gathered}$

That gives us two simultaneous equations in $a$ and $d$ Subtract the $u_{4}$ equation from the $u_{9}$ equation to eliminate $a$

$\begin{array}{rcl} 5 d & = 15 \\ d = 3 \end{array}$

Substitute that value into the first equation to find $a$

$\begin{gathered} a + 3 ( 3 ) & = 10 \\ a + 9 & = 10 \\ a = 1 \end{gathered}$

That is all the information we need to answer the question

$a=1,d=3$

Worked Example The sum of the first 10 terms of an arithmetic series is 630.

The first term is 18.

a) Find the common difference, $d$, of the series.

Use the arithmetic series formula $S_{n} = \frac{n}{2} [ 2 a + ( n - 1 ) d ]$

Here $n=10$, $S_{10}=630$ and $a=18$

Substitute in and solve for $d$

$\begin{gathered} 630 = \frac{10}{2} [ 2 ( 18 ) + ( 10 - 1 ) d ] \\ 630 = 5 [ 36 + 9 d ] \\ 630 = 180 + 45 d \\ 450 = 45 d \\ d = 450 \div 45 \end{gathered}$

$d=10$

The sum of the first 10 terms of another arithmetic series is also 630.

The common difference is 11.

b) Find the first term, $a$, of the series.

Here $n=10$, $S_{10}=630$ and $d=11$

$\begin{gathered} 630 = \frac{10}{2} [ 2 a + ( 10 - 1 ) 11 ] \\ 630 = 5 [ 2 a + 99 ] \\ 630 = 10 a + 495 \\ 135 = 10 a \\ a = 135 \div 10 \end{gathered}$

$a=13.5$