Everything about Geometric Sequence totally explained
In
mathematics, a
geometric progression, also known as a
geometric sequence, is a
sequence of
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 and 10, 5, 2.5, 1.25, ... is a geometric sequence with common ratio 1/2. The
sum of the terms of a geometric progression is known as a
geometric series.
Thus, the general form of a geometric sequence is
»
and that of a geometric series is
»
where
r ≠ 0 is the common ratio and
a is a
scale factor, equal to the sequence's start value.
Elementary properties
The
n-th term of a geometric sequence with initial value
a and common ratio
r is given by
» ,
which concludes the proof.
Relationship to geometry and Euclid's work
Books VIII and IX of
Euclid's
Elements analyze geometric progressions and give several of their properties.
A geometric progression gains its geometric character from the fact that the
areas of two
geometrically similar plane figures are in "duplicate" ratio to their corresponding sides; further the
volumes of two similar solid figures are in "triplicate" ratio of their corresponding sides.
The meaning of the words "duplicate" and "triplicate" in the previous paragraph is illustrated by the following examples. Given two squares whose sides have the ratio 2 to 3, then their areas will have the ratio 4 to 9; we can write this as 4 to 6 to 9 and notice that the ratios 4 to 6 and 6 to 9 both equal 2 to 3; so by using the side ratio 2 to 3 "in duplicate" we obtain the ratio 4 to 9 of the areas, and the sequence 4, 6, 9 is a geometric sequence with common ratio 3/2. Similarly, give two cubes whose side ratio is 2 to 5, their volume ratio is 8 to 125, which can be obtained as 8 to 20 to 50 to 125, the original ratio 2 to 5 "in triplicate", yielding a geometric sequence with common ration 5/2.
Elements, Book IX
The geometric progression 1, 2, 4, 8, 16, 32, ... (or, in the
binary numeral system, 1, 10, 100, 1000, 10000, 100000, ... ) is important in
number theory. Book IX, Proposition 36 of
Elements proves that if the sum of the first
n terms of this progression is a
prime number, then this sum times the
nth term is a
perfect number. For example, the sum of the first 5 terms of the series (1 + 2 + 4 + 8 + 16) is 31, which is a prime number. The sum 31 multiplied by 16 (the 5th term is the series) equals 496, which is a perfect number.
Book IX, Proposition 35 proves that in a geometric series if the first term is subtracted from the second and last term in the sequence then as the excess of the second is to the first, so will the excess of the last be to all of those before it. (This is a restatement of our formula for geometric series from above.) Applying this to the geometric progression 31,62,124,248,496 (which results from 1,2,4,8,16 by multiplying all terms by 31), we see that 62 minus 31 is to 31 as 496 minus 31 is to the sum of 31,62,124,248. Therefore the numbers 1,2,4,8,16,31,62,124,248 add up to 496 and further these
are all the numbers which
divide 496. For suppose that P divides
496 and it isn't amongst these numbers. Assume P×Q equals 16×31, or 31 is to Q as P is to 16. Now
P can't divide 16 or it would be amongst the numbers 1,2,4,8,16.
Therefore 31 can't divide Q. And since 31 doesn't divide Q and
Q measures 496, the
fundamental theorem of arithmetic implies that Q
must divide 16 and be amongst the numbers 1,2,4,8,16. Let Q be 4,
then P must be 124, which is impossible since by hypothesis P is not
amongst the numbers 1,2,4,8,16,31,62,124,248.
Further Information
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