Arithmetic Series

Arithmetic Series

A LevelAQAEdexcelOCREdexcel 2022OCR 2022

Arithmetic Series

A series is a sequence where the goal is to add all the terms together. We will study arithmetic series and geometric series.

Recall: Notation from Sequences:

a is first term

d is difference, the amount we add each time

n is the number of terms in the series

We will also introduce l, which is the last term of the series. Since there are n terms in the series and we have an nth term equation for arithmetic sequences, we have a formula for l:

l=a+(n-1)d

Make sure you are happy with the following topics before continuing.

A Level AQA Edexcel OCR

Adding the Terms

The sum of an arithmetic series with n terms is:

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

We can prove this result:

\begin{aligned}S_{n}&=a+(a+d)+(a+2d)+...+(l-d)+l\\[1.2em]&=l+(l-d)+(l-2d)+...+(a+d)+a\\[1.2em]2S_{n}&=(a+l)+(a+l)+(a+l)+...+(a+l)+(a+l)\\[1.2em]S_{n}&=\dfrac{n(a+l)}{2}\end{aligned}

 

If we substitute in our formula for l, we get:

S_{n}=\dfrac{n}{2}(2a+(n-1)d)

A LevelAQAEdexcelOCR

Sum Notation

\sum means sum, and we can use it instead of writing S_{n} to represent arithmetic series.

Example: \sum_{n=1}^{20}(3n+4) means the sum up to the 20th term of the arithmetic progression defined by 3n+4.

A LevelAQAEdexcelOCR

Natural Number Arithmetic Progressions

The sum of the first n natural numbers (positive whole numbers) is:

S_n = 1 + 2 + 3 + ... + (n-1) + n

So a=1, l = n and n = n.

If we put these values into the formula we have already seen, we would get:

S_n = \dfrac{1}{2} n (n+1)

A LevelAQAEdexcelOCR
A Level AQA Edexcel OCR

Example 1: Arithmetic Series in Practice

Jon is training for a marathon. Last week his furthest run was 4 miles. He plans to increase the length he runs by 1.1 miles per day. What is the total distance he has run by the time he reaches his goal of 26 miles?

[2 marks]

a=4

d=1.1

l=26

Find number of terms between a and l

n=\dfrac{26-4}{1.1}=\dfrac{22}{1.1}=20

Need to add 1 because both the first and last terms are included.

n=21

Substitute into formula:

\begin{aligned}S_{n}&=\dfrac{n(a+l)}{2} \\[1.2em] S_{21}&=\dfrac{21(4+26)}{2}\\[1.2em]&= \dfrac{21\times 30}{2}\\[1.2em]&=\dfrac{630}{2}\\[1.2em]&=315 \text{ miles} \end{aligned}

A LevelAQAEdexcelOCR

Example 2: Sum Notation

Find \sum_{n=1}^{50}(3n+4)

[2 marks]

This means the sum of the first 50 terms of the sequence defined by 3n+4.

a=3\times 1+4=3+4=7

d=3

n=50

Substitute into formula:

\begin{aligned}\sum_{n=1}^{50}(3n+4)&=\dfrac{50}{2}(2\times 7+(50-1)3)\\[1.2em]&=25(14+49\times 3)\\[1.2em]&=25(14+147)\\[1.2em]&=25\times 161\\[1.2em]&=4025\end{aligned}

A LevelAQAEdexcelOCR

Example Questions

i) \sum_{n=1}^{k}n

 

 

ii) Arithmetic progression with:

 

a=1

 

d=1

 

n=k

 

Sub into formula:

 

\begin{aligned}\sum_{n=1}^{k}n&=\dfrac{k}{2}(2\times 1+(k-1)1)\\[1.2em]&=\dfrac{k}{2}(2+k-1)\\[1.2em]&=\dfrac{k(k+1)}{2}\end{aligned}

 

 

iii) k=100

 

\begin{aligned}\sum_{n=1}^{100}n&=\dfrac{100(100+1)}{2}\\[1.2em]&=\dfrac{100\times 101}{2}\\[1.2em]&=\dfrac{10100}{2}\\[1.2em]&=5050\end{aligned}

 

 

iv) \sum_{n=1}^{k}n=55

 

\dfrac{k(k+1)}{2}=55

 

k(k+1)=110

 

k^{2}+k=110

 

k^{2}+k-110=0

 

(k-10)(k+11)=0

 

k=10 or k=-11

 

-11 not feasible

 

k=10

a=3

d=5

n=10

Substitute into formula:

\begin{aligned}S_{n}&=\dfrac{10}{2}(2\times 3+(10-1)5)\\[1.2em]&=5(6+9\times 5)\\[1.2em]&=5(6+45)\\[1.2em]&=5\times 51\\[1.2em]&=255\end{aligned}

2 years is 24 months.

a=10

d=10

n=24

Substitute into formula:

\begin{aligned}S_{n}&=\dfrac{24}{2}(2\times 10+(24-1)10)\\[1.2em]&=12(20+23\times 10)\\[1.2em]&=12(20+230)\\[1.2em]&=12\times 250\\[1.2em]&=3000\end{aligned}

S_{n}=184

 

a=9

 

d=4

 

Substitute into formula:

 

\dfrac{n}{2}(2\times 9+(n-1)4)=184

 

n(18+4n-4)=368

 

n(14+4n)=368

 

4n^{2}+14n=368

 

4n^{2}+14n-368=0

 

2n^{2}+7n-184=0

 

(n-8)(2n+23)=0

 

n=8 or n=-\dfrac{23}{2}

 

n=-\dfrac{23}{2} not feasible

 

n=8

\dfrac{1}{2}k (k+1) = 325

k^2 + k = 650

k^2 + k - 650 = 0

(k+26)(k-25) = 0

 

k cannot be negative so k = 25.

Related Topics

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Sequences

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Additional Resources

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Formula Booklet

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