Nth term and interpretation
The nth term is the final value in the requested finite sequence. The sum includes the initial term. A preview lists up to twelve terms so you can check the ratio and sign pattern.
To evaluate an individual real-number power, use the Exponent Calculator and follow its own input conventions.
The formula
Successive terms multiply by the supplied ratio. The implementation accumulates these finite terms directly, including the ratio-one case without dividing by zero.
Verified worked example
For first term 2, ratio 3, and five terms, the sequence is 2, 6, 18, 54, 162. Its final term is 162 and its finite sum is 242.
How to use this calculator
- Enter the initial term and the ratio used at every multiplication.
- Set a whole number of terms from one to one thousand.
- Check the final term and finite sum against the previewed pattern.
Geometric Sequence input conventions
A geometric sequence multiplies by a constant rather than adding a constant. Ratios between zero and one shrink a positive starting value; ratios above one enlarge it. Negative ratios alternate signs. A ratio of zero gives the first term followed by zeros, while a ratio of one repeats the first term. All of these finite patterns are handled without assigning a financial interpretation.
Interpreting nth term
The sum includes the starting term and ends at the requested index. This implementation accumulates the finite terms, avoiding division by a very small difference when the ratio is close to one. Very large progressions are rejected when they exceed finite numeric storage. Extremely small terms may underflow to zero; displayed rounding can also hide a small nonzero contribution.
Assumptions & limitations
What this calculation assumes
- Every successive term multiplies by the same supplied ratio; the sum is finite.
What to keep in mind
- Up to 1,000 terms; no symbolic expressions or infinite-series limits.
Common questions
Is this an infinite-series calculator?
No. The selected integer count always defines a finite sum. Convergence of an infinite geometric series requires an additional condition on the ratio.
Why does a negative ratio change signs?
Each multiplication by a negative number reverses the preceding sign. For example, a starting value of 2 with ratio −2 gives 2, −4, 8, −16.
What happens when the ratio is zero?
The first term remains as entered and every later term is zero. The finite sum is the first term regardless of how many terms are included.
Sources & further reading
Reviewed on