03 // Bidirectional STEM Calculators // MATHa_n = a₁ × r^(n - 1)

Geometric Sequence & Exponential Progression Calculator

A geometric progression multiplies each term by a constant non-zero common ratio r, generating exponential scaling governed by a_n = a₁ × r^(n - 1).

a_n = a₁ × r^(n - 1)

Pick which value you want to calculate (or type into any field):Solving for: a_n — N-th Geometric Term

N-th Geometric TermCalculated

Value of the progression at position n

0.00011.0000e+9

unit

First Term

Initial value at index n = 1

0.0011,000,000.00

unit

Common Ratio

Constant multiplicative factor between terms

0.0520

unit

Term Index Position

1-based index of the term

1100

unit

Calculated Result · N-th Geometric Term (a_n)

1,536.00

Step-by-Step Calculation

01Canonical Law: a_n = a₁ × r^(n - 1)

02Given Inputs (Base Units): a₁ = 3, r = 2, n = 10

03Solve for a_n (N-th Geometric Term): a_n = 1,536

A geometric progression multiplies each term by a constant non-zero common ratio r, generating exponential scaling governed by a_n = a₁ × r^(n - 1).

Finite Geometric Series SumS_n = a₁(1 - rⁿ) / (1 - r)

For a₁=3, r=2, n=10: S₁₀ = 3,069

Infinite Convergent Series (|r| < 1)S_∞ = a₁ / (1 - r)

Converges to a finite limit whenever -1 < r < 1

Geometric Mean of Adjacent Termsa_k = √(a_{k-1} × a_{k+1})

Logarithmic midpoint property

N-th Geometric Term (a_n) · First Term (a₁)

Compare across:

Geometric Sequence & Exponential Progression Calculator — First Term → N-th Geometric Term
#	First Term (a₁)	N-th Geometric Term (a_n)	Δ %
01	3 Current	1,536.00	0.00%
02	7,237.00	3,705,344.48	+241133.36%
03	14,474.00	7,410,688.45	+482366.70%
04	21,711.00	11,116,032.42	+723600.03%
05	28,948.00	14,821,376.39	+964833.36%
06	36,185.00	18,526,720.36	+1206066.69%
07	43,422.00	22,232,064.33	+1447300.02%
08	50,659.00	25,937,408.30	+1688533.35%
09	57,896.00	29,642,752.27	+1929766.68%
10	65,133.00	33,348,096.24	+2171000.02%
11	72,370.00	37,053,440.21	+2412233.35%
12	79,607.00	40,758,784.17	+2653466.68%
13	86,844.00	44,464,128.14	+2894700.01%
14	94,081.00	48,169,472.11	+3135933.34%
15	101,318.00	51,874,816.08	+3377166.67%
16	108,555.00	55,580,160.05	+3618400.00%
17	115,792.00	59,285,504.02	+3859633.33%
18	123,029.00	62,990,847.99	+4100866.67%
19	130,266.00	66,696,191.96	+4342100.00%
20	137,503.00	70,401,535.93	+4583333.33%

Was this tool helpful?

How do I find the common ratio r between two distant terms a₁ and a_n?+

Click SOLVE FOR on r to compute the (n - 1)-th root: r = (a_n / a₁)^(1 / (n - 1)).

When does an infinite geometric series have a finite sum?+

Whenever the absolute value of the common ratio is strictly less than 1 (|r| < 1), the infinite sum converges to S_∞ = a₁ / (1 - r).

How is compound interest related to a geometric sequence?+

Compound interest balances at year t form a geometric progression with common ratio r = (1 + rate / 100).