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Sequences and series
 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, . .
 Geometric series
 The sum of an infinite geometric sequence, infinite geometric series

An infinite geometric series converges (has a finite sum even when n is infinitely large) only if the absolute ratio of successive terms is less than 1 that is, if  -1 < r < 1. 
The sum of an infinite geometric series can be calculated as the value that the finite sum formula takes (approaches) as number of terms n tends to infinity,
first rewrite Sn, into
so that since | r | < 1, then rn ® 0 as  n ® oo
thus, the sum of an infinite converging geometric series.
 The sum of an infinite converging geometric series examples
 Example:  Given a square with side a. Its side is the diagonal of the second square. The side of this square is then the diagonal of the third square and so on, as shows the figure below. Find the sum of areas of all these squares.
 Solution:  Using the given conditions,
 Example:  Given an equilateral triangle with the side a. Its height is the side of another equilateral triangle. The height of this triangle is then the side of the third equilateral triangle and so on, as shows the figure below. Find the sum of areas of all these triangles.
 Solution:  a1 = h,  a2 = h1,  a3 = h2,  and so on. Thus,
 Converting recurring decimals (infinite decimals) to fraction
Recurring or repeating decimal is a rational number (fraction) whose representation as a decimal contains a pattern of digits that repeats indefinitely after decimal point.
The decimals that start their recurring cycle immediately after the decimal point are called purely recurring decimals. Purely recurring decimals convert to an irreducible fraction whose prime factors in the denominator can only be the prime numbers other than 2 or 5, i.e., the prime numbers from the sequence {3, 7, 11, 13,17, 19, . . }.
The decimals that have some extra digits before the repeating sequence of digits are called the mixed recurring decimals.
The repeating sequence may consist of just one digit or of any finite number of digits. The number of digits in the repeating pattern is called the period.
Mixed recurring decimals convert to an irreducible fraction whose denominator is a product of 2's and/or 5's besides the prime numbers from the sequence {3, 7, 11, 13,17, 19, . . . }.
All recurring decimals are infinite decimals.
 Converting purely recurring decimals to fraction
 Example:  Convert the purely recurring decimal to fraction.
 Solution:  Given decimal we can write as the sum of the infinite converging geometric series
Notice that, when converting a purely recurring decimal less than one to fraction, write the repeating digits to the numerator, and to the denominator of the equivalent fraction write as much 9's as is the number of digits in the repeating pattern.
Thus, for example:
 Converting mixed recurring decimals to fraction
 Example:  Convert the mixed recurring decimal to fraction.
 Solution:  Given decimal we can write as the sum of  0.3  and the infinite converging geometric series,
Since the repeating pattern is the infinite converging geometric series whose ratio of successive terms is less than 1, i.e., r = 0.01 then we use the formula for the sum of the infinite geometric series                    S¥ = a1 / (1 - r),
Notice that, when converting the mixed recurring decimal less than one to fraction, write the difference between the number formed by the entire sequence of digits, including the digits of the recurring part, and the number formed only by the digits of the non-recurring pattern to its numerator. To the denominator of the equivalent fraction write as much 9’s as is the number of digits in the repeating pattern and add as much 0’s as is the number of digits in the non-recurring pattern.
Thus, for example:
 
 
 
 
 
 
 
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