Sequence Calculator
Find the nth term and partial sum of an arithmetic or geometric sequence.
Work with either type of sequence: arithmetic (constant difference) or geometric (constant ratio).
How the Math Works
The Sequence Calculator uses two fundamental formulas to determine terms in sequences. For arithmetic sequences, the nth term is calculated using a + (n-1)d, where 'a' is the first term and 'd' is the common difference between terms. This linear relationship means each term increases or decreases by a fixed value. For geometric sequences, the nth term is found with a·r^(n-1), where 'r' is the common ratio. This exponential formula captures multiplicative patterns, where each term is a constant multiple of the previous one. Both formulas allow precise prediction of any term in the sequence, while the calculator also computes partial sums to determine cumulative totals up to a specific term.
Practical Applications
To apply this calculator, first identify whether your sequence follows an arithmetic or geometric pattern by checking if the difference or ratio between consecutive terms remains constant. Input the first term (a) and common difference (d) or ratio (r), then specify the term number (n) you wish to find. For example, to determine the 15th term in a geometric sequence starting at 2 with a ratio of 3, input a=2, r=3, n=15 to get 2·3^14. This tool is invaluable for analyzing patterns in finance, physics, or computer science where sequences model real-world phenomena like compound interest, motion, or algorithmic complexity.
Day-to-Day Use
In daily life, this calculator helps with practical tasks like projecting savings growth with regular deposits (arithmetic) or compound interest (geometric). For instance, if you save $50 weekly (arithmetic) or invest with 5% monthly returns (geometric), you can quickly find your total savings after a year. It also aids in understanding natural patterns, such as population growth or plant growth cycles, and simplifies planning for events with recurring intervals, like bus schedules or medication dosages, by predicting future occurrences accurately.
Worked example
Arithmetic 2, d = 3, n = 6 → term 17, sum 57.
FAQ
When does a geometric sum converge?
An infinite geometric series converges only when |r| < 1.