Arithmetic series (sum): formula, calculator and worked example

What the symbols mean

$S_{n}$: the sum of the first n terms, $S_{n}$
$a_{1}$: the first term
$d$: the common difference
$n$: how many terms are added

When to use it

Use it to total anything that grows by a fixed amount: savings that increase by the same sum each month, the seats in a theatre whose rows grow by two, the logs in a stacked pile. If you know the last term, the form $n/2(a₁ + aₙ)$ is quicker. Solving for n gives a quadratic; this calculator keeps only positive whole-number answers.

Worked example

Add up the whole numbers from 1 to 100 (a₁ = 1, d = 1, n = 100).

Start from the formula.
$S_{n} = \frac{n}{2} \cdot (2 a_{1} + (n - 1) d)$
Substitute $a_{1}$ = $1$ , $d$ = $1$ , $n$ = $100$ and work it out.
$S_{n} = \frac{100}{2} \cdot (2 \cdot 1 + (100 - 1) \cdot 1)$
$= 50 \cdot (2 + 99 \cdot 1)$
$= 50 \cdot (2 + 99)$
$= 50 \cdot 101$
$= 5050$
Check with the other form, n times the average of the first and last terms: the last term is $a_{n} = 100$ , and $\frac{100}{2} (1 + 100) = 5050$ .

Answer S₁₀₀ = 5050

Why it works

Write the sum forwards and backwards and add the two lines term by term: every pair adds to $a₁ + aₙ$ , and there are n pairs, so twice the sum is $n(a₁ + aₙ)$ . Substituting $aₙ = a₁ + (n − 1)d$ gives the formula.

Common mistakes

Using n where the formula has $n − 1$ inside the bracket.
Halving twice: the $n/2$ already takes care of the pairing, so do not divide the bracket by 2 as well.
Counting terms wrongly: from 5 to 50 in steps of 5 there are 10 terms, not 9 (use $n = (aₙ − a₁)/d + 1$ ).

Arithmetic series (sum)

Calculator

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What the symbols mean

When to use it

Worked example

Why it works

Common mistakes