Educational tool; results are estimates for learning, verify with your textbook or instructor.
This tool computes the $n$th term and the finite sum $S_n$ for an arithmetic sequence (common difference $d$) or a geometric sequence (common ratio $r$). You enter the first term, the step $d$ or $r$, and how many terms you want.
The output is meaningful when the list really is arithmetic or geometric, and when $n$ is a finite count of terms. A finite sum always exists as a number (unless it overflows). An infinite series is a different object: it is a limit of partial sums, and that limit may fail to exist.
This is not a convergence test and not a computer algebra system for general $\sum a_n$. It will not run a ratio test, a root test, or a comparison test. For those, read the series explainers.
How to use the closed forms
Choose Arithmetic or Geometric. Enter the first term $a_1$. For arithmetic, enter $d$ so that each term adds $d$. For geometric, enter $r$ so that each term multiplies by $r$. Enter $n\ge 1$ and press Compute.
The default arithmetic run is $2,5,8,\ldots$ with $n=10$. You should see $a_n=29$ and $S_n=155$. Switch to geometric with the default $r=0.5$ when you want a shrinking product.
How to read the output
The large line reports $a_n$ and $S_n$. For arithmetic, $a_n=a_1+(n-1)d$ and $S_n=\frac{n}{2}(2a_1+(n-1)d)$. For geometric, $a_n=a_1 r^{n-1}$ and $S_n=a_1(1-r^n)/(1-r)$ when $r\ne 1$. If $r=1$, the geometric sum is just $n\,a_1$.
$S_n$ is a partial sum, not a verdict about $\sum_{k=1}^\infty a_k$. A geometric series with $|r|<1$ does have a famous infinite sum $a_1/(1-r)$, and this page will not print that infinite value. Compute it by hand only after you know the infinite sum exists.
Common pitfalls
Large $|r|^n$ overflows double precision. The page will say so. That overflow is not a convergence test. It is a computer running out of range.
A common mistake is to treat a large finite $S_n$ as proof that the infinite series diverges, or a small $S_n$ as proof that it converges. Finite pictures can mislead. Use a test.
$n$ is a term index and a count of terms here, starting at $a_1$. If your course indexes from $a_0$, shift the formulas before you compare.
Related reading
The ideas live in sequences, series, and infinite sums and convergence tests for series. A Taylor polynomial is a different finite approximation: it matches a function and its derivatives at one point. The Taylor polynomial calculator builds that polynomial for e^x, sine, cosine, logarithms, and a few other standard functions. A telescoping series often starts by splitting a rational term. The partial fraction decomposition calculator writes that algebra. A sequence of partial sums settles only as a limit. The numerical limit estimator samples a function of a real variable. That is the same idea of approaching a value, and it is not a series test. Glossary: sequence, series, partial sum, convergence, ratio test, root test, and comparison test.
The surrounding map is calculus foundations. The limit of those partial sums, written as an argument, is real analysis and proof habits. A course path is the AP Calculus AB bridge.
