Arithmetic Progressions

Arithmetic Progressions (AP) An arithmetic progression is a list of numbers where the difference between successive numbers is constant. The terms in an arithmetic progression are usually denoted as T1 , T2 , and T3 , where T1 is the initial term in the progression, T2 is the second term, and so on. Thus, Tn is the nth term of the arithmetic progression. An example of an arithmetic progression is….
2; 4; 6; 8; 10; 12; 14;
Since the difference between successive terms is constant, we have….
T3 - T2 = T2 – T1 Thus, in general, we will denote the difference of the two consecutive arithmetic progression terms as “d”, which is a common notation.
Geometric Progressions (GP)
A geometric progression is a list of terms as in an arithmetic progression but in this case the ratio of successive terms is a constant. In other words, each term is a constant times the term that immediately precedes it. Let us write the terms in a geometric progression as T1 , T2 , T3 and so on. An example of a geometric progression is
10; 100; 1000; 10000;
Since the ratio of successive terms is constant, we have….

Thus, the ratio of successive terms is usually denoted by r and the first term again is usually written as T1. If we know the first term in a geometric progression and the ratio between successive terms, then we can work out the value of any term in the geometric progression. The formula to calculate the value of nth term is given by…

In certain cases, the sum of the terms in a geometric progression has a limit (note that this is summing together an infinite number of terms). A series like this has a limit partly because each successive term we are adding is smaller and smaller. When the sum of a geometric series has a limit, we say that exists and we can find the limit of the sum. The condition that exists is that r is greater than -1 but less than 1, i.e. . If this is the case, then we can use the formula for Sn above and let n grow arbitrarily big so that rn becomes as close as we like to zero. The formula to calculate is as follows…

Where “r” is the limit of the geometric progression so long as -1 < r < 1.