n-th Term
aₙ = a₁ × r^(n-1)
Sum of n Terms (r ≠ 1)
Sₙ = a₁ × (1 - rⁿ) / (1 - r)
Infinite Sum (|r| < 1)
S∞ = a₁ / (1 - r)
2, 6, 18, 54, ...
a₁ = 2, r = 3 (multiply by 3)
100, 50, 25, 12.5, ...
a₁ = 100, r = 0.5 (converges)
3, -6, 12, -24, ...
a₁ = 3, r = -2 (alternating)
A geometric sequence (or geometric progression) is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. For example, in the sequence 2, 6, 18, 54, each term is multiplied by 3 to get the next term, so the common ratio is 3.
Geometric sequences appear frequently in mathematics, science, and finance. They model phenomena like population growth, compound interest, radioactive decay, and signal amplification. Understanding geometric sequences is fundamental to calculus, particularly in the study of infinite series and convergence.
1. Enter the first term (a₁): This is the starting value of your sequence. It can be any real number, positive, negative, or decimal.
2. Enter the common ratio (r): This is the multiplier between consecutive terms. If r> 1, the sequence grows; if 0 < r < 1, it shrinks; if r < 0, the terms alternate in sign.
3. Enter the term number (n): Specify which term you want to find and how many terms to sum. The calculator will compute the n-th term and the sum of the first n terms.
4. Toggle options: Enable "Show steps" to see the detailed calculation process, and "Show sequence" to display the actual terms.
Geometric sequence calculations follow standard mathematical formulas. Results depend on correct input values and assumptions. For very large values of n or r, results may be subject to floating-point precision limitations.