Arithmetic Series (Summing an AP)
Learn arithmetic series for NSW Year 12 Mathematics Advanced (NESA outcome MAV-12-03). Adding up the terms of an arithmetic progression has a shortcut: the total equals the number of terms times the average of the first and last terms.
You will learn to use both sum formulas — one built from the first term and common difference, the other from the first and last terms — find how many terms are needed to reach a total, and solve problems about stacks, seating rows and steady savings plans.
Theory
An arithmetic series is the sum of the terms of an arithmetic sequence. This Year 12 Mathematics Advanced topic (MAV-12-03) uses \(S_n=\tfrac{n}{2}(a+\ell)\) and \(S_n=\tfrac{n}{2}[2a+(n-1)d]\) to find sums.
The sum of the first \(n\) terms is \(S_n\). There are two equivalent formulas: use \(S_n=\dfrac{n}{2}(a+\ell)\) when you know the last term \(\ell\), and \(S_n=\dfrac{n}{2}[\,2a+(n-1)d\,]\) when you know the common difference \(d\).
\(S_n\) is a quadratic function of \(n\). If given the last term, first find \(n\) with \(a_n=a+(n-1)d\).
Method
- Identify \(a\), and either the last term \(\ell\) or the difference \(d\).
- Find \(n\) first if you are given the last term, using \(a_n=a+(n-1)d\).
- Substitute into the matching sum formula.
| \(S_{20}\) | \(=\) | \(\dfrac{20}{2}[2(3)+19(4)]\) |
| \(=\) | \(10(82)=820\) |
| \(n\) | \(=\) | \(16\) |
| \(S_{16}\) | \(=\) | \(\dfrac{16}{2}(5+50)=440\) |
| \(S_{100}\) | \(=\) | \(\dfrac{100}{2}(1+100)\) |
| \(=\) | \(50\times101=5050\) |
| \(S_{10}\) | \(=\) | \(\dfrac{10}{2}[4+27]=155\) |
| \(n\) | \(=\) | \(10\) |
Common pitfalls
Frequently asked questions
What is the formula for the sum of an arithmetic series?
Use S n equals n over 2 times (a plus the last term), or S n equals n over 2 times (2a plus (n minus 1) d), where a is the first term and d the common difference.
Which sum formula should I use?
Use the first form, with a plus the last term, when you know the last term. Use the second form, with 2a plus (n minus 1) d, when you know the common difference.
How do you find the number of terms in an arithmetic series?
Use the nth term formula a n equals a plus (n minus 1) d, set it equal to the last term, and solve for n.
What is the sum of the first 100 whole numbers?
It is 100 over 2 times (1 plus 100), which equals 50 times 101, which is 5050.