Geometric Sequences and Series: Ratio, Sums and Growth

What is a geometric sequence?

A geometric sequence (or geometric progression) is a list of non-zero numbers a_1,a_2,a_3,… in which the ratio of each term to the one before it is always the same number r, called the common ratio. The rule is a_n+1=r a_n: to get the next term, multiply by r.

How to tell. Divide each term by the one before it, later over earlier. If every ratio is the same, the sequence is geometric. No term may be 0, because the ratio of a term to 0 is undefined and a 0 would make every later term 0.

Compare with arithmetic. An arithmetic sequence adds the same amount each step, so its terms lie on a straight line; a geometric sequence multiplies, so its terms lie on an exponential curve and eventually leave any arithmetic sequence far behind.

The geometric mean. For three neighbors of the same sign a_n^2=a_n-1a_n+1, so each term is the geometric mean √(ab) of the two beside it, as each term of an arithmetic sequence is their arithmetic mean. That is where the name comes from.

Type a list of numbers below and see whether it is geometric.

How do you find the n-th term of a geometric sequence?

The n-th term of a geometric sequence is a_n=a_1r^n-1: start at the first term and multiply by the ratio n-1 times. As in the arithmetic case, going from term 1 to term n takes n-1 steps.

For 3,6,12,… we have a_1=3 and r=2, so a_n=3·2^n-1 and a_10=3·2^9=1536. For 64,32,16,…, r=\frac12 and a_7=64·\left(\frac12\right)^6=1.

It is an exponential function. Writing a_n=(a_1)/(r) r^n shows the variable n in an exponent, so the laws of exponents apply, and the points of the sequence lie on a curve rather than a line. The size of r decides the picture:

• r>1: exponential growth. The terms get larger faster and faster.
• 0<r<1: exponential decay. The terms shrink toward 0 but never reach it.
• r<0: the signs alternate, and the sizes grow if |r|>1 or shrink if |r|<1.
• r=1: constant. r=-1: the sequence alternates between a_1 and -a_1.

Why it feels so fast. A penny, doubled each day for 30 days, is worth 2^29 cents on day 30, which is 5,368,709.12 dollars; adding a fixed 1,000 dollars a day would give only 30,000 dollars. Doubling beats any fixed addition in the end.

Which term is a given number? Solve a_1r^n-1=L. For 3,6,12,…,768: (768)/(3)=256=2^8, so n-1=8 and n=9. When the ratio is not a neat power, the number of terms needs logarithms, which are the inverse of the exponent.

Choose a first term, a ratio and a length below and watch the bars.

How do you find the ratio from two terms, and what is a geometric mean?

Two terms and their positions give the ratio: between term p and term q there are q-p multiplications by r, so a_q=a_p r^q-p and r^q-p=(a_q)/(a_p). Then a_1=a_p÷ r^p-1.

Example: a_2=6 and a_5=48. Then r^3=(48)/(6)=8, so r=2 and a_1=\frac62=3: the sequence is 3,6,12,24,48,…

How many answers? Taking a root can give two. If the number of steps q-p is odd there is one real ratio, with the sign of (a_q)/(a_p). If it is even there are two, +r and -r, provided (a_q)/(a_p) is positive: a_1=3 and a_3=12 fit both 3,6,12,… and 3,-6,12,…. If it is negative and the number of steps is even, no real ratio exists. And the ratio need not be a fraction: a_1=1 and a_3=2 give r=±√2, which is irrational.

Geometric means. Inserting k numbers between a and b so that the whole list is geometric is the same problem with k+1 steps: r^k+1=\frac ba. Two means between 3 and 81 have r^3=27, so r=3 and the means are 9 and 27. With a single mean, r^2=\frac ba and the mean is √(ab): between 2 and 32 it is √(64)=8.

Why the geometric mean matters. When a quantity grows by factors, the right average is the geometric one. Prices that rise by 10% and then 20% have grown by the factors 1.1 and 1.2, whose geometric mean is √(1.1·1.2)≈1.149, an average of about 14.9% a period, not the 15% that averaging the percentages would suggest.

Try two terms of your own below.

How do you find the sum of a geometric series?

The sum of the first n terms of a geometric sequence with r≠1 is S_n=(a_1(1-r^n))/(1-r), which is the same as (a_1(r^n-1))/(r-1); if r=1 every term is a_1 and S_n=na_1. A sum of terms like this is a geometric series.

Where the formula comes from. Multiply the sum by r and subtract; nearly everything cancels:

S_n=a_1+a_1r+a_1r^2+…+a_1r^n-1

rS_n= a_1r+a_1r^2+…+a_1r^n-1+a_1r^n

S_n-rS_n=a_1-a_1r^n ⇒ S_n(1-r)=a_1(1-r^n)

Dividing by 1-r is allowed only when r≠1, which is why that case is separate.

Examples.

• 3+6+12+… with 10 terms: S_10=(3(2^10-1))/(2-1)=3069.
• 1+\frac12+\frac14+…+\frac1512 with 10 terms: S_10=(1-(1/2)^10)/(1-1/2)=2-\frac1512=(1023)/(512).
• The Rhind Papyrus problem of seven houses with seven cats, seven mice, seven ears of wheat and seven measures of grain adds 7+49+343+2401+16807=19607, which is (7(7^5-1))/(6).
• The chessboard. One grain on the first square, two on the second, doubling on each of the 64 squares, gives 2^64-1=18,446,744,073,709,551,615 grains, far more than the world has grown.

A link with binary. With a_1=1 and r=2 the sum is 1+2+4+…+2^n-1=2^n-1, which is the number written as n ones in binary: 11111111_2=255=2^8-1, as in the article on binary numbers.

Watch the subtraction and the sum below, together with the partial sums.

When does an infinite geometric series have a sum?

An infinite geometric series has a sum exactly when |r|<1, and then S_∞=(a_1)/(1-r); for |r|\ge1 it diverges (unless every term is 0). The reason is in the finite formula: S_n=(a_1(1-r^n))/(1-r), and when |r|<1 the power r^n shrinks to 0, so S_n approaches (a_1)/(1-r). The gap after n terms is exactly (a_1r^n)/(1-r), which can be made as small as you like.

• \frac12+\frac14+\frac18+…=(1/2)/(1-1/2)=1: halving what is left of a unit square, over and over, fills the square.
• 6+4+\frac83+… has r=\frac23, so S_∞=(6)/(1/3)=18.
• 1-\frac12+\frac14-… has r=-\frac12, so S_∞=(1)/(3/2)=\frac23.
• 1+\frac14+\frac116+…=\frac43, the series Archimedes used for the area of a parabolic segment.

For |r|\ge1 the partial sums do not settle: r=1 gives na_1, which grows; r=-1 gives a_1,0,a_1,0,…; and |r|>1 grows without bound, as 1+2+4+8+… does. The formula (a_1)/(1-r) would give -1 for that last series, which is meaningless, so check |r|<1 before using it. An arithmetic series never converges, as the infinite arithmetic series section shows.

Repeating decimals are geometric series. 0.(3 repeating)=\frac310+\frac3100+… has a_1=\frac310 and r=\frac110, so it is (3/10)/(9/10)=\frac13. In the same way 0.(9 repeating)=(9/10)/(9/10)=1 exactly, and 0.(27 repeating)=(27/100)/(99/100)=(3)/(11). A repeating block of m digits has r=10^-m. This is the method behind the conversion of repeating decimals to fractions in the article on real numbers, and it is why every repeating decimal is rational.

Zeno's paradox. Achilles runs after a tortoise that has a head start. Each time he reaches where the tortoise was, it has moved a little further, so there are infinitely many steps. But the times of the steps form a geometric series with a finite sum, so Achilles does pass the tortoise, at a definite moment.

A bouncing ball. Dropped from 10 m and bouncing back to \frac35 of its previous height each time, a ball travels 10+2\left(6+3.6+2.16+…\right)=10+2·(6)/(1-3/5)=40 m in all.

Where do geometric sequences appear: interest, decay and growth?

Anything that changes by a fixed percentage each period is a geometric sequence: a rise of p\% multiplies by r=1+\frac p100 and a fall of p\% multiplies by r=1-\frac p100. Note that a 5% increase has ratio 1.05, not 0.05, and a 20% decrease has ratio 0.8, not 0.2.

Compound interest gives the balance P (1+i)^n after n periods at rate i per period. Interest that is added only to the original deposit instead gives an arithmetic sequence. Compounding more often helps only a little: the factor \left(1+\frac1n\right)^n approaches the irrational number e≈2.71828, which Jacob Bernoulli met in 1683 while studying exactly this, as the article on π, e and transcendental numbers recounts.

Decay never reaches zero. A half-life, a depreciating value or a cooling difference in temperature keeps getting multiplied by a ratio below 1, so it shrinks toward 0 without ever being 0.

Annuities are sums of geometric series. Each deposit grows by r=1+i for a different number of periods, so the total is P \bigl(1+(1+i)+…+(1+i)^n-1\bigr)=P ((1+i)^n-1)/(i), the sum formula with ratio 1+i.

Folding paper shows how fast doubling is. A sheet 0.1 mm thick folded 42 times, which cannot actually be done, would be 0.1·2^42 mm, about 440,000 km, more than the distance to the Moon.

How do you work with geometric sequences in Python and JavaScript?

**Build the terms with the power operator, a1 * r ** i in both languages, and use fractions.Fraction in Python when the sum must be exact.** Powers grow or shrink so fast that floats overflow to infinity or underflow to 0 long before the math becomes interesting.

>>> a1, r, n = 3, 2, 10
>>> [a1 * r ** i for i in range(5)]           # a_1 r^(i), i from 0
[3, 6, 12, 24, 48]
>>> a1 * r ** (n - 1)                         # the n-th term
1536
>>> a1 * (r ** n - 1) // (r - 1)              # the sum of n terms (exact for integers)
3069
>>> from fractions import Fraction
>>> sum(Fraction(1, 2) ** i for i in range(10))   # exact, no rounding
Fraction(1023, 512)
>>> 2 ** 64 - 1                               # the chessboard
18446744073709551615
>>> 1000 * 1.05 ** 10                         # compound interest, as a float
1628.894626777442
>>> 2.0 ** 1024                               # beyond the largest float: raises OverflowError

const terms = (a1, r, n) => Array.from({ length: n }, (_, i) => a1 * r ** i)
terms(3, 2, 5)                              // [3, 6, 12, 24, 48]
terms(3, 2, 10).reduce((s, x) => s + x, 0)  // 3069
2 ** 1024                                   // Infinity: beyond the largest float
0.5 ** 1075                                 // 0: smaller than the smallest float
2n ** 64n - 1n                              // 18446744073709551615n (BigInt is exact)

#include <iostream>
#include <cstdio>
#include <cmath>
using namespace std;

int main() {
  int a1 = 3, r = 2, n = 10;
  int term = a1, sum = 0, rn = 1;
  for (int i = 0; i < n; i++) {
    if (i < 5) cout << term << " ";                    // the first five terms
    sum += term;
    term *= r;
    rn *= r;
  }
  cout << endl;
  cout << a1 * (rn / r) << endl;                       // the n-th term: a1 r^(n-1)
  cout << sum << " " << a1 * (rn - 1) / (r - 1) << endl;   // the sum of n terms, by loop and by formula

  int num = 0;
  for (int i = 0; i < 10; i++) num += 1 << (9 - i);    // (1/2)^0 + ... + (1/2)^9 over the denominator 512
  cout << num << "/512" << endl;

  printf("%.0f\n", pow(2, 64) - 1);                    // 18446744073709551616: a double cannot hold 2^64 - 1
  printf("%.4f\n", 1000 * pow(1.05, 10));              // compound interest
  cout << (pow(2.0, 1024) > 1.7976931348623157e308) << endl;   // 1: beyond the largest double (inf)
  return 0;
}

/* output:
3 6 12 24 48
1536
3069 3069
1023/512
18446744073709551616
1628.8946
1
*/

The pitfalls, in order of how often they bite:

Exact fractions avoid the rounding of floats, as in the article on rational numbers. Check a closed form against a short loop before relying on it.

Where do geometric series come from?

Geometric progressions appear in the oldest mathematics, and the idea that infinitely many terms can add to a finite number took the Greeks centuries to accept.

• c. 1550 BCE. Problem 79 of the Rhind Papyrus lists seven houses, each with seven cats, each killing seven mice, and so on: the powers of 7 and their sum, 19607.
• 5th century BCE. Zeno's paradoxes of motion, such as Achilles and the tortoise, puzzled the Greeks because they seemed to add infinitely many steps.
• c. 300 BCE. Euclid's Elements, Book IX, proposition 35, gives the sum of terms in geometric progression in the language of proportions, the first proof of the finite sum formula.
• c. 250 BCE. Archimedes, in the Quadrature of the Parabola, summed 1+\frac14+\frac116+…=\frac43 to find the area of a parabolic segment, an early use of an infinite series with the sum found by a careful argument.
• 1683. Jacob Bernoulli found the number e as the limit of compound interest \left(1+\frac1n\right)^n.

A well-known legend tells of the inventor of chess who asked a king for one grain of rice on the first square, two on the second and so on, and asked for more than the kingdom could ever supply. The tale is old, with versions told in Arabic sources by the 13th century, and it remains the most memorable way to feel how fast 2^n grows.

What are the common mistakes with geometric sequences?

The most common mistakes with geometric sequences are the nine below, each with the correct statement and a number that shows why.

Practice: test your understanding

These questions mix everything above. A wrong answer costs nothing here: read the hint and try again.

Summary: geometric sequences and series at a glance

• Definition: (a_n+1)/(a_n)=r is constant and no term is 0; each term is the geometric mean of its neighbors.
• n-th term: a_n=a_1r^n-1, exponential in n; |r|>1 grows, |r|<1 decays, r<0 alternates.
• From two terms: r^q-p=(a_q)/(a_p) (one or two real ratios); k means between a and b: r^k+1=\frac ba.
• Sum of n terms: S_n=(a_1(1-r^n))/(1-r) for r≠1, and na_1 for r=1.
• Infinite series: S_∞=(a_1)/(1-r) only for |r|<1; repeating decimals are such series.
• Percent change: a rise of p\% has r=1+\frac p100, a fall has r=1-\frac p100; compound interest is P(1+i)^n.
• Code: a1 * r ** (n - 1); use integers, BigInt or Fraction to avoid float overflow and rounding.

Glossary: key terms

Geometric sequence

A sequence of non-zero numbers in which each term is the previous one times the same fixed number, such as 3, 6, 12, 24.

Common ratio

The constant multiplier r between neighboring terms of a geometric sequence, found by dividing a term by the one before it.

Geometric series

The sum of the terms of a geometric sequence, finite or infinite.

Exponential growth

Growth in which a quantity is multiplied by the same factor greater than 1 in each period, as in a geometric sequence with ratio above 1.

Exponential decay

Decrease in which a quantity is multiplied by the same factor between 0 and 1 in each period, so it shrinks toward 0 but never reaches it.

Geometric mean

For two positive numbers a and b, the number equal to the square root of a times b; each term of a geometric sequence is the geometric mean of its neighbors.

Partial sum

The sum S sub n of the first n terms of a series.

Convergent series

A series whose partial sums approach a fixed finite number, called its sum; an infinite geometric series converges exactly when the ratio is smaller than 1 in size.

Divergent series

A series whose partial sums do not approach a finite number, such as 1 plus 2 plus 4 plus 8 and so on.

Sum to infinity

The value a convergent infinite series approaches, equal to the first term over 1 minus the ratio for a geometric series.

Compound interest

Interest that is added to the balance so that it earns interest itself, which makes the balance a geometric sequence with ratio 1 plus the rate.

Half-life

The time in which a decaying quantity falls to half its value, so that it is multiplied by one half each half-life.

Annuity

A series of equal payments at regular intervals, whose total value is a geometric sum.

Arithmetic sequence

A sequence in which each term is the previous one plus the same fixed number, as opposed to multiplied by it.

Frequently asked questions

References

1. James Stewart, Lothar Redlin and Saleem Watson (2016). Precalculus: Mathematics for Calculus (7th ed.), chapter 12 (sequences and series). Cengage Learning.
2. Euclid. Elements, Book IX, proposition 35 (the sum of numbers in geometric progression). c. 300 BCE.
3. Archimedes. Quadrature of the Parabola. c. 250 BCE.
4. Arnold Buffum Chace (1927). The Rhind Mathematical Papyrus. Mathematical Association of America.
5. Jacob Bernoulli (1690). Quaestiones nonnullae de usuris, cum solutione problematis de sorte alearum. Acta Eruditorum.
6. Walter Rudin (1976). Principles of Mathematical Analysis (3rd ed.), chapter 3 (series). McGraw-Hill.
7. Ronald Graham, Donald Knuth and Oren Patashnik (1994). Concrete Mathematics: A Foundation for Computer Science (2nd ed.), chapter 2 (sums). Addison-Wesley.
8. Python Software Foundation. The Python Standard Library: fractions, rational numbers. docs.python.org. https://docs.python.org/3/library/fractions.html.

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