Finite arithmetic series Grade 12
Table of Contents
Finite arithmetic series
Finite arithmetic series ,An arithmetic sequence is a sequence of numbers, such that the difference between any term and the previous term is a constant number called the common difference (d
): thus Finite arithmetic series Grade 12
where
 Tn
is the nth
 term of the sequence;
 a
 is the first term;
 d
 is the common difference.
When we sum a finite number of terms in an arithmetic sequence, we get a finite arithmetic series.
The sum of the first one hundred integers
A simple arithmetic sequence is when a=1
and d=1
, which is the sequence of positive integers:
If we wish to sum this sequence from n=1
to any positive integer, for example 100
, we would write
This gives the answer to the sum of the first 100
positive integers.
The mathematician, Karl Friedrich Gauss, discovered the following proof when he was only 8 years old. His teacher had decided to give his class a problem which would distract them for the entire day by asking them to add all the numbers from 1
to 100. Young Karl quickly realised how to do this and shocked the teacher with the correct answer, 5Â 050
. This is the method that he used:
 Write the numbers in ascending order.
 Write the numbers in descending order.
 Add the corresponding pairs of terms together.
 Simplify the equation by making Sn
 the subject of the equation.
General formula for a finite arithmetic series (EMCDY)
If we sum an arithmetic sequence, it takes a long time to work it out termbyterm. We therefore derive the general formula for evaluating a finite arithmetic series. We start with the general formula for an arithmetic sequence of n terms and sum it from the first term (a) to the last term in the sequence (l
):
This general formula is useful if the last term in the series is known.
We substitute l=a+(nâˆ’1)d
into the above formula and simplify:
The general formula for determining the sum of an arithmetic series is given by:
or
For example, we can calculate the sum S20
for the arithmetic sequence Tn=3+7(nâˆ’1)
by summing all the individual terms:
or, more sensibly, we could use the general formula for determining an arithmetic series by substituting a=3
, d=7 and n=20
:
This example demonstrates how useful the general formula for determining an arithmetic series is, especially when the series has a large number of terms.
Worked example 7: General formula for the sum of an arithmetic sequence
Find the sum of the first 30
terms of an arithmetic series with Tn=7nâˆ’5
by using the formula.
Use the general formula to generate terms of the sequence and write down the known variables
This gives the sequence: 2;9;16â€¦
Write down the general formula and substitute the known values
Write the final answer
S30=3105
Worked example 8: Sum of an arithmetic sequence if first and last terms are known
Find the sum of the series âˆ’5âˆ’3âˆ’1+â‹¯â‹¯+123
Identify the type of series and write down the known variables
Determine the value of n
Use the general formula to find the sum of the series
Write the final answer
S65=3835
Worked example 9: Finding n
given the sum of an arithmetic sequence
Given an arithmetic sequence with T2=7
and d=3, determine how many terms must be added together to give a sum of 2Â 146
.
Write down the known variables
Use the general formula to determine the value of n
but n
must be a positive integer, therefore n=37.
We could have solved for n
using the quadratic formula but factorising by inspection is usually the quickest method.
Write the final answer
S37=2146
Worked example 10: Finding n
given the sum of an arithmetic sequence
The sum of the second and third terms of an arithmetic sequence is equal to zero and the sum of the first 36
terms of the series is equal to 1Â 152
. Find the first three terms in the series.
Write down the given information
Solve the two equations simultaneously
Write the final answer
The first three terms of the series are:
Calculating the value of a term given the sum of n terms:
If the first term in a series is T1
, then S1=T1.
We also know the sum of the first two terms S2=T1+T2
, which we rearrange to make T2the subject of the equation:
Similarly, we could determine the third and fourth term in a series:
Tn=Snâˆ’Snâˆ’1,Â forÂ nâˆˆ{2;3;4;â€¦}
and T1=S1

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Interactive Exercises