QuickCalculators finds the nth term and partial sum for arithmetic, geometric and Fibonacci sequences, listing every term up to n in a chip row. The number sequence calculator switches closed-form geometric sums to a times n when the common ratio equals one instead of dividing by zero.
Find the nth term of an arithmetic sequence
Use a_n equals a_1 plus (n minus 1) times d. First term 2 with difference 5 yields term 20 at n equals 20 as 2 plus 19 times 5 equals 97. The chip list shows 2, 7, 12, 17 along the way.
Find the nth term of a geometric sequence
Use a_n equals a times r to the (n minus 1). Ratio two on first term three gives term 6 as 3 times 2 to the 5 equals 96. Negative ratios alternate sign term to term.
Generate a Fibonacci sequence
Fibonacci mode lists terms with F₁ = 1 and F₂ = 1, then each later term as the sum of the two before it, using BigInt once values grow past ordinary integers. The tenth term is 55.
Consecutive-term ratios approach the golden ratio near 1.618, which is why standalone fibonacci queries redirect into this sequence page rather than a separate tool.
Each new Fibonacci term is the sum of the two before it, so the chip row grows as 0, 1, 1, 2, 3, 5, 8, 13, and onward. Consecutive ratios approach the golden ratio near 1.618, which QuickCalculators can surface beside later terms when the sequence is long enough for that ratio to settle.
Sum the terms of a sequence
Arithmetic partial sum uses n times (a_1 plus a_n) over 2. Geometric partial sum uses a times (1 minus r to the n) over (1 minus r) when r is not one. Fibonacci sums accumulate term by term in the work list.
Tell a sequence from a series
A sequence lists terms; a series sums them. The nth term line answers sequence questions; the sum line answers series questions. Labels on outputs keep the vocabulary straight for exam wording.
When the geometric sum formula breaks
If common ratio r equals one, every term equals the first term a and the sum of n terms is a times n, not the closed form with (1 minus r) in the denominator. The engine detects r equals 1 and applies the linear sum rule.
Avoid this common mistake
Students plug r equals 1 into (1 minus r^n) over (1 minus r) and divide by zero. A geometric sequence with ratio one is constant; sum n copies of a as a times n. The calculator branches automatically when r equals one.
Frequently asked questions
What is an arithmetic sequence?
An arithmetic sequence adds a fixed difference d each step. The number sequence calculator uses a_n equals a_1 plus (n minus 1) times d.
What is the formula for the nth term of a geometric sequence?
The formula for the nth term of a geometric sequence is a_n equals a times r to the (n minus 1). Enter first term and ratio in geometric mode.
What is the 20th term of the sequence 2, 7, 12, 17?
The 20th term of 2, 7, 12, 17 is 97 because the difference is 5 and 2 plus 19 times 5 equals 97. Use arithmetic mode with a_1 equals 2 and d equals 5.
How do you find the sum of an arithmetic sequence?
To find the sum of an arithmetic sequence, use n times (first plus last) over 2. The tool computes the last term before summing when you request n terms.
What are the first ten Fibonacci numbers?
The first ten Fibonacci numbers with F₁ = 1 and F₂ = 1 are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. Fibonacci mode lists them through the chip row and reports the nth term on the primary line.
What is the difference between a sequence and a series?
A sequence is an ordered list of terms; a series is the accumulation of those terms. This page reports both nth term and partial sum outputs separately.
Why does the geometric sum formula fail when the ratio is 1?
The geometric sum formula fails when the ratio is 1 because the denominator (1 minus r) becomes zero. The sum reduces to a times n instead.
Summary
QuickCalculators handles arithmetic, geometric and Fibonacci sequences with nth terms, partial sums, and listed terms through n. Geometric sums use a times n when r equals one to avoid a zero denominator. Pick mode, enter parameters, read term and sum lines together for exam-style sequence and series questions.