Sequences and Series Geometric sequence/progression The sum of the first n terms of a finite geometric sequence, geometric series
Geometric sequences, examples Recursive definition and the recursion formula
Geometric sequence or progression
A sequence of numbers in which the ratio r of each two successive terms an / an -1 is constant or whose each term is r times the preceding.
For example, in the sequence 1, 3, 9, 27, . . . ,  each term is 3 times the preceding, that is r = 3.
Therefore, a geometric sequence can be written as
a1a2a3a4, . . . , an -1an, . . .    or    a1a1 · ra1 · r2a1 · r3, . . . , an -1an, . . .
where,    a2 = a1 · r,
a3 = a1 · r2,
a4 = a1 · r3, and so on.
So, the formula for the nth term, or the general term of a geometric sequence is
 an = a1 · rn -1.
A geometric sequence is said to be convergent if   -1 < r < 1 that is, an approaches zero as n becomes infinitely large thus, the limit of the sequence is 0.
An infinite sequence that has no a finite limit is called a divergent sequence.
The sum of the first n terms of a finite geometric sequence, geometric series
The sum of numbers in a geometric sequence we call the geometric series and write,
Sn = a1 + a2 + a3 + . . . + an -1 + an,
that is         Sn = a1 + a1 · r + a1 · r2 + . . . + a1 · rn -2 + a1 · rn -1,
or         Sn = a1 · (1 + r + r2 + . . . +  rn -2 + rn -1)
 thus, the sum of the first n terms of a finite geometric sequence or series.
Recall that by factoring binomial   xn - yn = (x - y) · (xn -1 + xn -2y + xn -3y2 + . . . + xyn -2 + y n -1)
so that,    rn - 1n = (r - 1) · (rn -1 + rn -2 + rn -3 + . . . + r2 + r + 1).
Or, we can use following method to derive the same formula,
Sn = a1 + a1 · r + a1 · r2 + . . . + a1 · rn -2 + a1 · rn -1,
and           r ·  Sn = a1 · r + a1 · r2 + a1 · r3 + . . . + a1 · rn -1 + a1 · rn,
then    r · Sn - Sn = a1 · rn - a1   or    Sn · (r - 1) = a1 · (rn - 1)  =>   Sn = a1 · (rn - 1) / (r - 1).
Geometric sequences, examples
Example:  Find the sum of the first six terms of the geometric sequence 1, 3, 9, 27, 81, 243, 729, . . .
Solution:   Since,   a1 = 1,  r = 3  and  n = 6
we plug these values into the formula for the first n terms of a finite geometric sequence,
Sn = [a1 · (rn - 1)] / (r - 1) = [1 · (36 - 1)] / (3 - 1) = 728 / 2 = 364.
Example:  Find the first term a1 and the number of terms n in a geometric sequence with the general term an = 192, the sum of the first n terms Sn = 381 and the common ratio r = 2.
Solution:  Using the formulas for an and Sn we get the system of two equations in two unknown, Therefore, the geometric sequence is  3, 6, 12, 24, 48, 96, 192 . . .
Example:  Write the geometric sequence such that the sum of its first three terms is 21 and the difference between the first and the second term is 12.
Solution:  Given are the equations,
(1)  a1 + a2 + a3 = 21      or       a1 + a1r + a1r2 = 21,          a1 · (1 + r + r2) = 21
and    (2)  a1 - a2 = 12                        a1 - a1r = 12,                    a1 · (1 - r) = 12 Thus, the two geometric sequences satisfy the given conditions,
(a1)1 = 3    and   r1 = -3    give    3, -9, 27
(a1)2 = 16  and   r2 = 1/4   give   16, 4, 1.
Example:  In a geometric sequence the first term a1 = 3, the general term an = 729 and the sum
Sn = 1092. Find the common ratio r and the number of terms n.
Solution:  Plug the given values into the formulas for an and Sn, (1)  rn = 243r     <=   r = 3
3n = 243 · 3,     3n = 729,     3n = 36   =>    n = 6
The geometric sequence is  3, 9, 27, 81, 243, 729, 2187, . . .
Example:  Find the first term a1, the common ratio r and the number of terms n of a geometric sequence if given  a6 - a4 = 24 a4 + a3 = 12 and Sn = 511.
Solution:      (1)  a6 - a4 = 24,         a1r5 - a1r3 = 24,         a1r3 · (r2 - 1) = 24
(2)  a4 + a3 = 12,         a1r3 + a1r2 = 12,          a1r2 · (r + 1) = 12 The geometric sequence is  1, 2, 4, 8, 16, 32, 64, 128, 256, . . .
Example:   In a geometric sequence, the difference between the fifth and the third term relates to the difference between the third and the second term as 12 : 1. The sum of the first three terms is 65. Find the geometric sequence.
Solution:      (1)  (a5 - a3) : (a3 - a2= 12 : 1   or     (a1r4 - a1r2) / (a1r2 - a1r) = 12, Thus, the two geometric sequences satisfy the given conditions,
a1 = 5    and   r1 = -4     give    5, -20, 80, -320, 1280, -5120, . . .
a1 = 5    and   r2 = 3       give    5, 15, 45, 135, 405, 1215, . . .
Recursive definition and the recursion formula
A recursion formula is the part of a recursive definition.
Recursive definition is the definition of a sequence by specifying its first term and the pattern or algorithm by which each term of the sequence is generated from the preceding.
That is, a recursion formula shows how each term of the sequence relates to the preceding term.
For example, given the first term
a1 = 3    and the recursion formula,    an = an - 1 + 4,  where   an - an - 1 = d,    d = 4
specifies the successive terms of the arithmetic sequence  3, 7, 11, 15, 19, 23, 27, . . .
Example:  Write the first five terms of the sequence if,  a1 = 7  and  an = 2an - 1 - 3.
Solution: Multiply the preceding term by 2 and subtract 3, obtained is the sequence 7, 11, 19, 35, 67, . .   Intermediate algebra contents 