Sum of an Arithmetic Series

Statement

An arithmetic series is the sum of the terms in an arithmetic sequence, where each term increases by a fixed difference \(d\). The formula for the sum of the first \(n\) terms is:

\[ S_n = \frac{n}{2}\left[2a + (n-1)d\] = \frac{n}{2}(a_1 + a_n) \]

Here, \(a\) (or \(a_1\)) is the first term, \(d\) is the common difference, \(a_n\) is the \(n\)th term, and \(n\) is the number of terms.

Why it’s true

• Pairing method: Writing the series forwards and backwards shows each pair sums to the same total.
• For example: \(1+2+3+\cdots+100\). Writing it backwards: \(100+99+98+\cdots+1\). Adding both gives \(101+101+101+\cdots\) (100 times).
• So, total sum = \(n \times (a_1 + a_n)/2\).

Recipe (how to use it)

1. Identify the first term \(a\), the common difference \(d\), and the number of terms \(n\).
2. Either use \(S_n = \tfrac{n}{2}[2a+(n-1)d]\) directly, or find the last term \(a_n\) and use \(S_n = \tfrac{n}{2}(a_1+a_n)\).
3. Substitute values and simplify.

Spotting it

These appear whenever the question asks for the sum of a sequence with a constant step, e.g. “Find the total of the first 50 even numbers.”

Common pairings

• Formula for the nth term of an arithmetic sequence: \(a_n = a + (n-1)d\).
• Word problems involving money, steps, or repeated patterns.

Mini examples

1. Given: Find the sum of the first 10 terms of \(2, 5, 8, 11, …\). Answer: \(S_{10} = 10/2 × (2+29) = 155\).
2. Given: Find the sum of the first 20 natural numbers. Answer: \(S_{20} = 20/2 × (1+20) = 210\).

Pitfalls

• Forgetting to divide by 2 at the end.
• Mixing up nth term formula with sum formula.
• Arithmetic slips when simplifying.

Exam strategy

• Write out first few terms to check the sequence is arithmetic.
• Use the version of the formula that fits the given information.
• Always check your answer is reasonable — sums should be much larger than individual terms.

Summary

The sum of an arithmetic series can be found using the formula \(S_n = \frac{n}{2}[2a+(n-1)d]\) or \(S_n = \frac{n}{2}(a_1+a_n)\). Both methods use the idea of pairing terms from the start and end of the sequence.