v Natural numbers are the product of human spirit. – DEDEKIND v . Introduction In mathematics, the word, “ sequence ” is used in much the same way as it is in ordinary English. When we say that a collection of objects is listed in a sequence, we usually mean that the collection is ordered in such a way that it has an identified first member, second member, third member and so on.
For example, population of human beings or bacteria at different times form a sequence. The amount of money deposited in a bank, over a number of years form a sequence. Depreciated values of certain commodity occur in a sequence. Sequences have important applications in several spheres of human activities.
Sequences, following specific patterns are called progressions . In previous class, we have studied about arithmetic progression (A.P). In this Chapter, besides discussing more about A.P.; arithmetic mean, geometric mean, relationship between A.M. and G.M., special series in forms of sum to n terms of consecutive natural numbers, sum to n terms of squares of natural numbers and sum to n terms of cubes of natural numbers will also be studied.
. Sequences Let us consider the following examples: Assume that there is a generation gap of years, we are asked to find the number of ancestors, i.e., parents, grandparents, great grandparents, etc. that a person might have over years. Here, the total number of generations = = Fibonacci ( - )