How this instrument works
An arithmetic sequence adds the same fixed difference at every step (3, 8, 13, 18, ...), while a geometric sequence multiplies by the same fixed ratio instead (2, 6, 18, 54, ...) — two of the most common sequence patterns, growing at fundamentally different rates: linearly for arithmetic, exponentially for geometric. This page finds the nth term of either kind from the same three starting pieces of information: the first term, the common difference or ratio, and which term number is wanted.
The arithmetic formula, aₙ = a₁+(n−1)d, adds the difference (n−1) times to reach the nth term, since the first term itself already counts as the starting point with zero additions applied. The geometric formula, aₙ = a₁·dⁿ⁻¹, works the same way but with multiplication standing in for addition, since geometric growth compounds rather than accumulates.
The type selector lets both formulas live on one page specifically because they're so often confused with each other — a sequence described as 'growing by 5 each time' is arithmetic, while one described as 'growing by a factor of 5 each time' is geometric, and mixing up which growth pattern applies is one of the most common early mistakes with sequences.
- Enter the sequence type into the Sequence type field: 1 for arithmetic, 2 for geometric.
- Enter the first term into the First term field.
- Enter the common difference (arithmetic) or ratio (geometric) into the matching field.
- Enter which term number is wanted into the Term number (n) field.
- Read nth term: the sheet applies the matching formula automatically.
Worked example — arithmetic, starting at 3, difference 5
An arithmetic sequence starts at 3 with a common difference of 5: 3, 8, 13, 18. The 4th term is 3+(4−1)×5=18 — a straight, linear climb, adding the same fixed amount at every step.
Switching to geometric with the same starting values (first term 2, ratio 3): 2, 6, 18, 54 — the 4th term is 2×3³=54, a much faster, exponential climb, since each step multiplies rather than adds.
Questions
What is the difference between arithmetic and geometric sequences?
An arithmetic sequence adds the same fixed difference at every step, growing linearly. A geometric sequence multiplies by the same fixed ratio instead, growing exponentially — a fundamentally faster growth pattern once the ratio exceeds 1.
How do you find the nth term of an arithmetic sequence?
aₙ = a₁ + (n−1)d, where a₁ is the first term, d is the common difference, and n is the term number — the difference gets added exactly (n−1) times to reach the nth term from the first.
How do you find the nth term of a geometric sequence?
aₙ = a₁ · dⁿ⁻¹, where d here is the common ratio — the same structure as the arithmetic formula, but with multiplication and an exponent standing in for addition.
What does a negative common difference or ratio mean?
For an arithmetic sequence, a negative difference means the sequence steadily decreases. For a geometric sequence, a negative ratio means the sequence alternates sign at every step, in addition to growing or shrinking in magnitude.
Can the term number be a fraction?
No — term numbers are meant to be whole numbers (1st term, 2nd term, and so on), since a sequence is fundamentally a discrete, one-at-a-time listing rather than a continuous function.