Geometric Progression Calculator
Calculate nth term, sum, and properties of geometric sequences and series. Enter values for instant results with step-by-step formulas.
Formula
a_n = a * r^(n-1) | S_n = a(r^n - 1)/(r - 1) | S_inf = a/(1-r)
Where a_n = nth term, a = first term, r = common ratio, n = number of terms, S_n = sum of first n terms (for r not equal to 1), and S_inf = sum of infinite series (only when |r| < 1). The nth term formula multiplies the first term by the ratio raised to the (n-1) power.
Worked Examples
Example 1: Compound Investment Growth
Problem: An investment of $5,000 grows at 8% annually. Find the value after 10 years and the total growth factor. Also calculate the sum of all year-end values.
Solution: First term a = 5000, Common ratio r = 1.08, n = 10\n10th term (value after 10 years): a_10 = 5000 x 1.08^9 = 5000 x 1.999 = $9,995.02\nNote: after 10 FULL years, value = 5000 x 1.08^10 = $10,794.62\nSum of year-end values (years 1-10):\nS_10 = 5000(1.08^10 - 1)/(1.08 - 1)\n= 5000(2.1589 - 1)/0.08\n= 5000 x 14.487 = $72,432.81\nTotal growth factor: 1.08^10 = 2.159
Result: Year 10 value: $10,794.62 | Sum of all values: $72,432.81 | Growth: 2.16x
Example 2: Bouncing Ball Distance
Problem: A ball is dropped from 10 meters and each bounce reaches 60% of the previous height. Find the total distance traveled before it stops.
Solution: The ball falls 10m, bounces up 6m, falls 6m, bounces up 3.6m, etc.\nDownward distances: 10, 6, 3.6, 2.16, ... (GP with a=10, r=0.6)\nUpward distances: 6, 3.6, 2.16, ... (GP with a=6, r=0.6)\nTotal down = 10/(1-0.6) = 10/0.4 = 25 meters\nTotal up = 6/(1-0.6) = 6/0.4 = 15 meters\nTotal distance = 25 + 15 = 40 meters\nAlternatively: Total = 10 + 2(6)/(1-0.6) = 10 + 30 = 40 meters
Result: Total distance: 40 meters | Infinite bounces converge to finite distance
Frequently Asked Questions
What is a geometric progression and what defines it?
A geometric progression (GP), also called a geometric sequence, is an ordered list of numbers where each term after the first is obtained by multiplying the previous term by a fixed nonzero number called the common ratio, denoted by r. If the first term is a and the common ratio is r, the sequence is a, ar, ar^2, ar^3, and so on. For example, 3, 6, 12, 24, 48 is a geometric progression with first term 3 and common ratio 2. Unlike arithmetic progressions where terms grow linearly, geometric progressions exhibit exponential growth or decay. When the common ratio is between -1 and 1 (exclusive), the terms decrease in absolute value toward zero.
How do you find the nth term of a geometric progression?
The nth term of a geometric progression is given by the formula a_n = a times r^(n-1), where a is the first term, r is the common ratio, and n is the position number. This formula works because each step multiplies by r, so reaching the nth term requires (n-1) multiplications by r starting from a. For example, in the GP 2, 6, 18, 54, ..., the 8th term is 2 times 3^7 = 2 times 2187 = 4374. Geometric progressions grow or shrink much faster than arithmetic ones: a GP with ratio 2 doubles every step, reaching over a million by the 20th term from just 1. This exponential behavior makes GPs essential for modeling compound growth phenomena.
What is the formula for the sum of a finite geometric series?
The sum of the first n terms of a geometric series is S_n = a(r^n - 1)/(r - 1) when r is not equal to 1, or S_n = na when r equals 1. This formula is derived by multiplying the series by r, subtracting the original from the result, and solving for S. For example, the sum of the first 6 terms of the GP 3, 6, 12, 24, 48, 96 is S_6 = 3(2^6 - 1)/(2 - 1) = 3(64 - 1)/1 = 3 times 63 = 189. An equivalent form is S_n = a(1 - r^n)/(1 - r), which is more convenient when |r| < 1. The formula works for any real common ratio except r = 1, and the choice of which form to use depends on whether r is greater or less than 1.
When does an infinite geometric series converge and what is its sum?
An infinite geometric series converges (has a finite sum) if and only if the absolute value of the common ratio is less than 1, that is |r| < 1. When this condition is met, the infinite sum is S = a / (1 - r). As n approaches infinity, r^n approaches zero when |r| < 1, causing the partial sum formula S_n = a(1 - r^n)/(1 - r) to approach a/(1 - r). For example, the infinite series 1 + 1/2 + 1/4 + 1/8 + ... has sum = 1/(1 - 0.5) = 2. When |r| >= 1, the terms do not decrease toward zero, so the series diverges. This convergence property has profound applications in mathematics, physics, and finance, from calculating present values of perpetuities to analyzing bouncing ball distances.
What is the geometric mean and how does it relate to geometric progressions?
The geometric mean of two positive numbers p and q is sqrt(p times q), and it is the value that, when placed between p and q, forms a three-term geometric progression with equal ratios. More generally, inserting k geometric means between two positive numbers a and b creates a GP of k+2 terms, where the common ratio r = (b/a)^(1/(k+1)). For example, inserting 2 geometric means between 2 and 54 gives r = (54/2)^(1/3) = 27^(1/3) = 3, producing the sequence 2, 6, 18, 54. In a GP, every term (except the first and last) equals the geometric mean of its two neighbors. The geometric mean is always less than or equal to the arithmetic mean for positive numbers, with equality only when both numbers are the same.
What are common real-world applications of geometric progressions?
Geometric progressions model any phenomenon involving constant percentage change over equal intervals. Compound interest is the most classic example: an investment growing at 5% annually follows a GP with ratio 1.05. Population growth and decay, including bacterial growth and radioactive decay, follow geometric patterns. In music, the frequencies of notes in an equal-tempered scale form a GP with ratio 2^(1/12) because each semitone multiplies the frequency by this constant. Computer science uses GPs in algorithm analysis, where doubling input size has multiplicative effects on runtime. Signal attenuation in telecommunications, drug concentration decay in pharmacology, and mortgage amortization all involve geometric progressions.