Arithmetic and geometric are particular sequences which are used to solve problems involving number sequences which have a special pattern. With arithmetic sequences, the pattern is that the same number, called the common difference, is always added to the proceeding one, as in:3, 7,11, 15, 19, . . .
With geometric sequences, the pattern is that the same number, called the common ratio, is always multiplied to the proceeding number, as in:3, 6, 12, 24, 48, . . .
Because of these constant patterns, much is known about each type of sequence.
Any term of an arithmetic sequence with its common difference, denoted by d, can be predicted using an explicit formula, since= + d
= + d = ( + d) + d = + 2 d
and so forth - each time adding one more "d" to obtain the next number.
So the explicit formula to generate any term of the sequence becomes:= + ( n - 1 ) * d
Any term of an geometric sequence with its common ratio, denoted by r, can be predicted using an explicit formula, since= * r
= * r = ( * r ) * r = *
and so forth - each time multiplying one more "r" to obtain the next number.
So the explicit formula to generate any term of the sequence becomes:= *
Both the arithmetic and geometric sequences have what are called sum formulas, which allow us to obtain the sum of the first n terms of either sequence.
The sum of the first n terms of an arithmetic sequence can be found by:
while the sum of the first n terms of a geometric sequence can be found by:
The derivations of these formulas are on pages 258 (arithmetic) and 259 (geometric) in the text.
Thomas Malthus in 1798 said: "that population, when unchecked, increased in a geometric ration, and subsistence for man in an arithmetical ratio." How does this work out? Suppose we assume that:
- 1) the population is growing at a rate of 2% per year
- 2) it takes 1 acre to provide food for one person
- 3) the world has 8 billion acres of arable land
- 4) the world population is 1975 was 4 billion,
How long would it be before the world reaches maximum population?
Since the population is growing at the rate of 2%, the common ratio would be 1.02. The question then becomes for what value of n would:
Using a calculator, approximately, when n = 35. So adding the 35 to 1975 = 2010 which is when the world would reach maximum population which would also mean massive starvation because of distribution problems.