Sequences and Series, Let Us Consider the Following Problems, Objectives, Sequence

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  • Succession of numbers of which one number is designated as the first, other as the second, and another as the third and so on gives rise to what is called a sequence. Sequences have wide applications.

  • In this lesson we shall discuss particular types of sequences called arithmetic sequence, geometric sequence and also find arithmetic mean (A.M), geometric mean (G.M) between two given numbers.

  • We will also establish the relation between A.M and G.M.

Let Us Consider the Following Problems

A man places a pair of newly born rabbits into a warren and wants to know how many rabbits he would have over a certain period of time. A pair of rabbits will start producing off springs two months after they were born and every following month one new pair of rabbits will appear. At the beginning the man will have in his warren only one pair of rabbits, during the second month he will have the same pair of rabbits, during the third month the number of pairs of rabbits in the warren will grow to two; during the fourth month there will be three pairs of rabbits in the warren. Thus, the number of pairs of rabbits in the consecutive months are:

The recurring decimal can be written as a sum

A man earns Rs. on the first day, Rs. on the second day, Rs. on the third day and so on. The day to day earning of the man may be written as

We may ask what his earnings will be on the day in a specific month.

Again let us consider the following sequences:

Objectives

After studying this lesson, you will be able to:

  • Describe the concept of a sequence (progression);

  • Define an A.P. and cite examples;

  • Find common difference and general term of a A.P;

  • Find the fourth quantity of an A.P. given any three of the quantities , , and ;

  • Calculate the common difference or any other term of the A.P. given any two terms of the A.P;

  • Derive the formula for the sum of ‘’ terms of an A.P;

  • Calculate the fourth quantity of an A.P. given three of , , and ;

  • Insert A.M. between two numbers;

  • Solve problems of daily life using concept of an A.P;

  • State that a geometric progression is a sequence increasing or decreasing by a definite multiple of a non-zero number other than one;

  • Identify G.P.’s from a given set of progressions;

  • Find the common ratio and general term of a G.P;

  • Calculate the fourth quantity of a G.P when any three of the quantities , , and are given;

  • Calculate the common ratio and any term when two of the terms of the G.P. are given;

  • Write progression when the general term is given;

  • Derive the formula for sum of n terms of a G.P;

  • Calculate the fourth quantity of a G.P. if any three of , , and are given;

  • Derive the formula for sum of infinite number of terms of a G.P. when ;

  • Find the third quantity when any two of , and are given;

  • Convert recurring decimals to fractions using G.P;

  • Insert G.M. between two numbers;

  • Establish relationship between A.M. and G.M.

Sequence

A sequence is a collection of numbers specified in a definite order by some assigned law, whereby a definite number of the set can be associated with the corresponding positive integer . The different notations used for a sequence are.

Let us consider the following sequences:

In the above examples, the expression for nth term of the sequences are as given below:

For all positive integer .

Also for the first problem in the introduction, the terms can be obtained from the relation

A finite sequence has a finite number of terms. An infinite sequence contains an infinite number of terms.

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