Geometric Series Calculator

A series is geometric when each term is the previous one times a fixed common ratio $r$:

$a + ar + ar^2 + \cdots = \sum_{n=1}^{N} ar^{n-1}$

Finite sum of the first $n$ terms, valid for any $r \neq 1$:

$S_n = a\,\frac{1 - r^n}{1 - r} = a\,\frac{r^n - 1}{r - 1}$

Infinite sum, valid only when $|r| < 1$:

$S_\infty = \frac{a}{1 - r}$

The restriction is not decoration. If $|r| \ge 1$ the terms never shrink to zero, $r^n$ does not vanish, and the series diverges — there is no sum to find. When $r = 1$ the series is just $a$ added $n$ times, so $S_n = na$.

Here $a$ is the first term of the series you are actually summing, not necessarily the first term of the underlying sequence.

Step by step

1. Find $a$: the first term you are told to include.
2. Find $r$: divide any term by the one before it, $r = \dfrac{a_{n+1}}{a_n}$. Test two different pairs — if they disagree, the series is not geometric and none of these formulas apply.
3. Count $n$: the number of terms, not the last index. Summing from $k = 0$ to $k = 5$ is six terms.
4. Choose the formula: finite $S_n$ for a fixed count, $S_\infty$ only after checking $|r| < 1$.
5. Substitute and simplify, keeping exact fractions where possible.

Reading sigma notation

$\sum_{k=0}^{n-1} ar^{k} \quad\text{and}\quad \sum_{k=1}^{n} ar^{k-1}$

are the same series written two ways. Substitute the lower limit into the general term to read off $a$ directly — that removes the off-by-one error entirely.

Working backwards

Given two terms $a_p$ and $a_q$, the ratio follows from $r^{q-p} = a_q / a_p$, and then $a = a_p / r^{\,p-1}$.

• Using $S_\infty$ without checking $|r| < 1$: with $r = 2$ the formula returns $-a$, a confidently wrong finite number for a divergent series.
• Off-by-one in $n$: $\sum_{k=0}^{5}$ has six terms. Count them, or convert the index so the sum starts at $1$.
• Confusing sequence and series: the $n$-th term is $ar^{n-1}$; the sum of the first $n$ terms is $S_n$. "Find the sum of the sequence" means the series.
• Dropping the sign of a negative ratio: for $r = -\frac{2}{5}$, $1 - r = 1 + \frac{2}{5}$. Alternating series still converge whenever $|r| < 1$.
• Mixing in an arithmetic pattern: constant differences mean arithmetic, constant ratios mean geometric. Check by division, not subtraction.