What is a Sequence Calculator?
This calculator computes terms and sums for arithmetic and geometric sequences.
How it's calculated
Arithmetic: aₙ = a₁ + (n−1)d, sum Sₙ = n(a₁+aₙ)/2
Geometric: aₙ = a₁rⁿ⁻¹, sum Sₙ = a₁(1−rⁿ)/(1−r)
Step-by-step example
Arithmetic sequence with first term 3, common difference 4 — find the 10th term and the sum of the first 10 terms
- Step 1 10th term: a₁₀ = 3 + (10−1)×4 = 39
- Step 2 Apply the sum formula: S₁₀ = 10×(3+39)÷2
- Step 3 S₁₀ = 10×42÷2 = 210
Interpretation: This is exactly Gauss's trick — pair the first and last terms, then multiply by half the count.
Young Gauss and the sum trick
As a schoolboy, Gauss was reportedly asked to add all the numbers from 1 to 100 — and solved it in seconds. His insight: pairing the numbers from opposite ends gives 1+100=101, 2+99=101, and so on, with 50 identical pairs summing to 101 each. That gives 101 × 50 = 5,050, which is exactly the formula Sₙ = n(first term + last term)/2. He didn't memorize a formula — he saw the underlying structure.
Good to know
- Compound interest is a geometric sequence — the same percentage multiplies the principal each period, which is why compound growth is sometimes described as 'the eighth wonder of the world.'
- The Rule of 72 comes from this: divide 72 by the interest rate to estimate how many years it takes for an investment to double. At 6% annual interest, that's about 12 years.
- The classic chessboard-and-rice-grains story is a geometric sequence in action: doubling grains on each square leads to over 18 quintillion grains by the 64th square — far exceeding total world rice production.
- The Fibonacci sequence (1, 1, 2, 3, 5, 8, 13...) has consecutive terms whose ratio converges toward the golden ratio (1.618…), observed in sunflower seed patterns and pinecone spirals.
- A geometric series only converges to a finite sum when the common ratio's absolute value is less than 1 — this same principle proves that 0.9999… equals exactly 1.
Frequently asked questions
Q. How do I tell arithmetic from geometric?
If consecutive terms have a constant difference, it's arithmetic. If they have a constant ratio, it's geometric.
Q. Does the sum go to infinity if the ratio is greater than 1?
Yes, if there are infinitely many terms, it diverges. A finite number of terms can still be summed with the formula.
Q. What kind of sequence describes compound interest?
Geometric — the balance is multiplied by (1 + interest rate) every period.