Diagram illustrating three basic geometric sequences of the pattern 1(rn−1) up to 6 iterations deep. The first block is a unit block and the dashed line represents the infinite sum of the sequence, a number that it will forever approach but never touch: 2, 3/2, and 4/3 respectively.
In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with common ratio 3. Similarly 10, 5, 2.5, 1.25, ... is a geometric sequence with common ratio 1/2.
Examples of a geometric sequence are powersrk of a fixed non-zero number r, such as 2k and 3k. The general form of a geometric sequence is
where r ≠ 0 is the common ratio and a ≠ 0 is a scale factor, equal to the sequence's start value.
The distinction between a progression and a series is that a progression is a sequence, whereas a series is a sum.
The n-th term of a geometric sequence with initial value a = a1 and common ratio r is given by
−1, the absolute value of each term in the sequence is constant and terms alternate in sign.
less than −1, for the absolute values there is exponential growth towards (unsigned) infinity, due to the alternating sign.
Geometric sequences (with common ratio not equal to −1, 1 or 0) show exponential growth or exponential decay, as opposed to the linear growth (or decline) of an arithmetic progression such as 4, 15, 26, 37, 48, … (with common difference 11). This result was taken by T.R. Malthus as the mathematical foundation of his Principle of Population.
Note that the two kinds of progression are related: exponentiating each term of an arithmetic progression yields a geometric progression, while taking the logarithm of each term in a geometric progression with a positive common ratio yields an arithmetic progression.
An interesting result of the definition of the geometric progression is that any three consecutive terms a, b and c will satisfy the following equation:
where b is considered to be the geometric mean between a and c.
5 × 312 )
(1 − 5) × 312
Computation of the sum 2 + 10 + 50 + 250. The sequence is multiplied term by term by 5, and then subtracted from the original sequence. Two terms remain: the first term, a, and the term one beyond the last, or arm. The desired result, 312, is found by subtracting these two terms and dividing by 1 − 5.
A geometric series is the sum of the numbers in a geometric progression. For example:
Letting a be the first term (here 2), n be the number of terms (here 4), and r be the constant that each term is multiplied by to get the next term (here 5), the sum is given by:
In the example above, this gives:
The formula works for any real numbers a and r (except r = 1, which results in a division by zero). For example:
Since the derivation (below) does not depend on a and r being real, it holds for complex numbers as well.
To derive this formula, first write a general geometric series as:
We can find a simpler formula for this sum by multiplying both sides
of the above equation by 1 − r, and we'll see that
since all the other terms cancel. If r ≠ 1, we can rearrange the above to get the convenient formula for a geometric series that computes the sum of n terms:
If one were to begin the sum not from k=1, but from a different value, say m, then
provided . If then the sum is of just the constant and so equals .
Differentiating this formula with respect to r allows us to arrive at formulae for sums of the form
For a geometric series containing only even powers of r multiply by 1 − r2 :
Equivalently, take r2 as the common ratio and use the standard formulation.
An infinite geometric series is an infinite series whose successive terms have a common ratio. Such a series converges if and only if the absolute value of the common ratio is less than one (|r| < 1). Its value can then be computed from the finite sum formula
Animation, showing convergence of partial sums of geometric progression (red line) to its sum (blue line) for .
Diagram showing the geometric series 1 + 1/2 + 1/4 + 1/8 + ⋯ which converges to 2.
For a series containing only even powers of ,
and for odd powers only,
In cases where the sum does not start at k = 0,
The formulae given above are valid only for |r| < 1. The latter formula is valid in every Banach algebra, as long as the norm of r is less than one, and also in the field of p-adic numbers if |r|p < 1. As in the case for a finite sum, we can differentiate to calculate formulae for related sums.
This formula only works for |r| < 1 as well. From this, it follows that, for |r| < 1,
It is a geometric series whose first term is 1/2 and whose common ratio is −1/2, so its sum is
The summation formula for geometric series remains valid even when the common ratio is a complex number. In this case the condition that the absolute value of r be less than 1 becomes that the modulus of r be less than 1. It is possible to calculate the sums of some non-obvious geometric series. For example, consider the proposition
The proof of this comes from the fact that
which is a consequence of Euler's formula. Substituting this into the original series gives
This is the difference of two geometric series, and so it is a straightforward application of the formula for infinite geometric series that completes the proof.
The product of a geometric progression is the product of all terms. It can be quickly computed by taking the geometric mean of the progression's first and last individual terms, and raising that mean to the power given by the number of terms. (This is very similar to the formula for the sum of terms of an arithmetic sequence: take the arithmetic mean of the first and last individual terms, and multiply by the number of terms.)
As the geometric mean of two numbers equals the square root of their product, the product of a geometric progression is:
(An interesting aspect of this formula is that, even though it involves taking the square root of a potentially-odd power of a potentially-negative r, it cannot produce a complex result if neither a nor r has an imaginary part. It is possible, should r be negative and n be odd, for the square root to be taken of a negative intermediate result, causing a subsequent intermediate result to be an imaginary number. However, an imaginary intermediate formed in that way will soon afterwards be raised to the power of , which must be an even number because n by itself was odd; thus, the final result of the calculation may plausibly be an odd number, but it could never be an imaginary one.)
Let P represent the product. By definition, one calculates it by explicitly multiplying each individual term together. Written out in full,
Carrying out the multiplications and gathering like terms,
The exponent of r is the sum of an arithmetic sequence. Substituting the formula for that calculation,
^"Set Partitions: Stirling Numbers". Digital Library of Mathematical Functions. Retrieved 24 May 2018.
^Friberg, Jöran (2007). "MS 3047: An Old Sumerian Metro-Mathematical Table Text". In Friberg, Jöran (ed.). A remarkable collection of Babylonian mathematical texts. Sources and Studies in the History of Mathematics and Physical Sciences. New York: Springer. pp. 150–153. doi:10.1007/978-0-387-48977-3. ISBN 978-0-387-34543-7. MR 2333050.
^Heath, Thomas L. (1956). The Thirteen Books of Euclid's Elements (2nd ed. [Facsimile. Original publication: Cambridge University Press, 1925] ed.). New York: Dover Publications.
Hall & Knight, Higher Algebra, p. 39, ISBN 81-8116-000-2