Geometric Progression Calculator
Calculate nth term, sum, and properties of geometric sequences and series. Enter values for instant results with step-by-step formulas.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Geometric Progression Calculator
Calculator
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Formula: a_n = a * r^(n-1) | S_n = a(r^n - 1)/(r - 1) | S_inf = a/(1-r)
Worked example — Year 10 value: $10,794.62 | Sum of all values: $72,432.81 | Growth: 2.16x
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 10th term (value after 10 years): a_10 = 5000 x 1.08^9 = 5000 x 1.999 = $9,995.02 Note: after 10 FULL years, value = 5000 x 1.08^10 = $10,794.62 Sum of year-end values (years 1-10): S_10 = 5000(1.08^10 - 1)/(1.08 - 1) = 5000(2.1589 - 1)/0.08 = 5000 x 14.487 = $72,432.81 Total 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. Downward distances: 10, 6, 3.6, 2.16, ... (GP with a=10, r=0.6) Upward distances: 6, 3.6, 2.16, ... (GP with a=6, r=0.6) Total down = 10/(1-0.6) = 10/0.4 = 25 meters Total up = 6/(1-0.6) = 6/0.4 = 15 meters Total distance = 25 + 15 = 40 meters Alternatively: 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.
How do you find the common ratio when given two terms?
When given two terms at known positions in a geometric progression, the common ratio can be found using r = (a_m / a_n)^(1/(m-n)), where a_m and a_n are the terms at positions m and n respectively. For example, if the 3rd term is 12 and the 7th term is 192, then r = (192/12)^(1/(7-3)) = 16^(1/4) = 2. Once r is found, the first term can be calculated as a = a_n / r^(n-1). In our example, a = 12 / 2^2 = 12/4 = 3. When working with negative terms or even differences in position, there may be multiple valid ratios (including negative ones), so context must guide which solution is physically meaningful.
What happens when the common ratio is negative?
When the common ratio is negative, the terms of the geometric progression alternate in sign, creating an oscillating sequence. For example, with a = 3 and r = -2, the sequence is 3, -6, 12, -24, 48, -96, and so on, where each term has the opposite sign of its predecessor and double the absolute value. The finite sum formula S_n = a(r^n - 1)/(r - 1) still applies. For convergence of the infinite series, the condition |r| < 1 still holds, meaning r must be between -1 and 1 (exclusive). The alternating series 1 - 1/2 + 1/4 - 1/8 + ... with r = -1/2 converges to 1/(1-(-0.5)) = 1/1.5 = 2/3. Negative ratios arise in oscillating physical systems and alternating current analysis.
How is the product of terms in a geometric progression calculated?
The product of the first n terms of a GP with first term a and common ratio r is P_n = a^n times r^(n(n-1)/2). This formula arises because the product is a times ar times ar^2 times ... times ar^(n-1) = a^n times r^(0+1+2+...+(n-1)) = a^n times r^(n(n-1)/2), using the sum formula for integers. For example, the product of the first 4 terms of the GP 2, 6, 18, 54 is 2^4 times 3^(4 times 3/2) = 16 times 3^6 = 16 times 729 = 11664. This can be verified: 2 times 6 times 18 times 54 = 11664. An elegant alternative formula uses the geometric mean: P_n = (geometric mean)^n, where the geometric mean of the first and last terms equals sqrt(a times a_n).
What is the relationship between geometric series and fractional representations?
Geometric series provide an elegant method for converting repeating decimals to fractions, revealing the deep connection between these mathematical concepts. Any repeating decimal can be expressed as an infinite geometric series with |r| < 1. For example, 0.333... = 3/10 + 3/100 + 3/1000 + ... is a GP with a = 3/10 and r = 1/10, summing to (3/10)/(1 - 1/10) = (3/10)/(9/10) = 1/3. Similarly, 0.142857142857... = 142857/1000000 + 142857/1000000000000 + ... sums to 142857/999999 = 1/7. This technique proves that every repeating decimal is rational. The formula S = a/(1-r) essentially performs this conversion automatically, making geometric series the bridge between decimal and fractional representations of rational numbers.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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