`star` Introduction

`star` Sequences

`star` Series

`star` Sequences

`star` Series

`\color{red} ✍️` In mathematics, the word, `color(blue)"“sequence”"` is used in much the same way as it is in ordinary English.

`\color{red} ✍️` 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.

`\color{red} ✍️` 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.

`\color{fuchsia} \(★ mathbf\ul"finite sequence")`

A sequence containing finite number of terms is called a finite sequence.

For example, sequence of ancestors is a finite sequence since it contains `10` terms (a fixed number).

`\color{fuchsia} \(★ mathbf\ul"infinite sequence")`

A sequence is called infinite, if it is not a finite sequence. For example, the sequence of successive quotients mentioned above is an infinite sequence, infinite in the sense that it never ends.

`color(red)"Let us consider the following examples:"`

`color(red)(=>)` Assume that there is a generation gap of `30` years, we are asked to find the number of ancestors, i.e., parents, grandparents, great grandparents, etc. that a person might have over `300` years.

Here, the total number of generations `= 300/30= 10`

The number of person’s ancestors for the first, second, third, …, tenth generations are 2, 4, 8, 16, 32, …, 1024. These numbers form what we call `color(blue)"a sequence."`

`color(red)(=>)` Consider the successive quotients that we obtain in the division of `10` by `3` at different steps of division.

In this process we get `3,3.3,3.33,3.333, ... `and so on. These quotients also form a sequence.

`\color{red} ✍️` The various numbers occurring in a sequence are called `color(blue)"its terms."` We denote the terms of a sequence by `a_1, a_2, a_3, …, a_n, …,` etc., the subscripts denote the position of the term.

`\color{red} ✍️` The `color(blue)(n^(th))` term is the number at the `n^(th)` position of the sequence and is denoted by `color(blue)(a_n)`.

`\color{red} ✍️` The `n^(th)` term is also called `color(blue)"the general term of the sequence."`

Thus, the terms of the sequence of person’s ancestors mentioned above are:

` \ \ \ \ \ \ \ \ \ \ \ \ a_1 = 2, a_2 = 4, a_3 = 8, …, a_10 = 1024.`

Similarly, in the example of successive quotients

` \ \ \ \ \ \ \ \ \ \ \ \ a_1 = 3, a_2 = 3.3, a_3 = 3.33, …, a_6 = 3.33333,` etc.

`\color{fuchsia} \(★ mathbf\ul"Fibonacci sequence")`

In some cases, an arrangement of numbers such as `1, 1, 2, 3, 5, 8,..` has no visible pattern, but the sequence is generated by the recurrence relation given by

` \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \a_1 = a_2 = 1`

` \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \a_3 = a_1 + a_2`

` \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \a_n = a_(n – 2) + a_(n – 1), n > 2`

This sequence is called `color(blue)"Fibonacci sequence."` we use the functional notation `a(n)` for `a_n.`

Often, it is possible to express the rule, which yields the various terms of a sequence in terms of algebraic formula.

`color(red)(=>)` Consider for instance, the sequence of even natural numbers `2, 4, 6, …`

Here `a_1 = 2 = 2 × 1 \ \ \ \ \ a_2 = 4 = 2 × 2`

`\ \ \ \ \ a_3 = 6 = 2 × 3 \ \ \ \ \ \ a_4 = 8 = 2 × 4`

`.... .... .... .... .... ....`

`.... .... .... .... .... ....`

`a_(23) = 46 = 2 × 23, \ \ \ \ a_(24) = 48 = 2 × 24, ` and so on.

In fact, we see that the `n^(th)` term of this sequence can be written as `color(red)(a_n = 2n),` where `n` is a natural number.

`color(red)(=>)` Similarly, in the sequence of odd natural numbers `color(red)(1,3,5, ......…,)`

The `n^(th)` term is given by the formula, `color(red)(a_n = 2n – 1),` where `n` is a natural number.

a sequence can be regarded as a function whose domain is the set of natural numbers or some subset of it of the type {1, 2, 3...k}. Sometimes, we use the functional notation a(n) for `a_n`.

A sequence containing finite number of terms is called a finite sequence.

For example, sequence of ancestors is a finite sequence since it contains `10` terms (a fixed number).

`\color{fuchsia} \(★ mathbf\ul"infinite sequence")`

A sequence is called infinite, if it is not a finite sequence. For example, the sequence of successive quotients mentioned above is an infinite sequence, infinite in the sense that it never ends.

`color(red)"Let us consider the following examples:"`

`color(red)(=>)` Assume that there is a generation gap of `30` years, we are asked to find the number of ancestors, i.e., parents, grandparents, great grandparents, etc. that a person might have over `300` years.

Here, the total number of generations `= 300/30= 10`

The number of person’s ancestors for the first, second, third, …, tenth generations are 2, 4, 8, 16, 32, …, 1024. These numbers form what we call `color(blue)"a sequence."`

`color(red)(=>)` Consider the successive quotients that we obtain in the division of `10` by `3` at different steps of division.

In this process we get `3,3.3,3.33,3.333, ... `and so on. These quotients also form a sequence.

`\color{red} ✍️` The various numbers occurring in a sequence are called `color(blue)"its terms."` We denote the terms of a sequence by `a_1, a_2, a_3, …, a_n, …,` etc., the subscripts denote the position of the term.

`\color{red} ✍️` The `color(blue)(n^(th))` term is the number at the `n^(th)` position of the sequence and is denoted by `color(blue)(a_n)`.

`\color{red} ✍️` The `n^(th)` term is also called `color(blue)"the general term of the sequence."`

Thus, the terms of the sequence of person’s ancestors mentioned above are:

` \ \ \ \ \ \ \ \ \ \ \ \ a_1 = 2, a_2 = 4, a_3 = 8, …, a_10 = 1024.`

Similarly, in the example of successive quotients

` \ \ \ \ \ \ \ \ \ \ \ \ a_1 = 3, a_2 = 3.3, a_3 = 3.33, …, a_6 = 3.33333,` etc.

`\color{fuchsia} \(★ mathbf\ul"Fibonacci sequence")`

In some cases, an arrangement of numbers such as `1, 1, 2, 3, 5, 8,..` has no visible pattern, but the sequence is generated by the recurrence relation given by

` \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \a_1 = a_2 = 1`

` \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \a_3 = a_1 + a_2`

` \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \a_n = a_(n – 2) + a_(n – 1), n > 2`

This sequence is called `color(blue)"Fibonacci sequence."` we use the functional notation `a(n)` for `a_n.`

Often, it is possible to express the rule, which yields the various terms of a sequence in terms of algebraic formula.

`color(red)(=>)` Consider for instance, the sequence of even natural numbers `2, 4, 6, …`

Here `a_1 = 2 = 2 × 1 \ \ \ \ \ a_2 = 4 = 2 × 2`

`\ \ \ \ \ a_3 = 6 = 2 × 3 \ \ \ \ \ \ a_4 = 8 = 2 × 4`

`.... .... .... .... .... ....`

`.... .... .... .... .... ....`

`a_(23) = 46 = 2 × 23, \ \ \ \ a_(24) = 48 = 2 × 24, ` and so on.

In fact, we see that the `n^(th)` term of this sequence can be written as `color(red)(a_n = 2n),` where `n` is a natural number.

`color(red)(=>)` Similarly, in the sequence of odd natural numbers `color(red)(1,3,5, ......…,)`

The `n^(th)` term is given by the formula, `color(red)(a_n = 2n – 1),` where `n` is a natural number.

a sequence can be regarded as a function whose domain is the set of natural numbers or some subset of it of the type {1, 2, 3...k}. Sometimes, we use the functional notation a(n) for `a_n`.

`\color{red} ✍️` Let `color(red)(a_1, a_2, a_3,…..............,a_n,)` be a given `color(blue)(sequence)`.

`\color{red} ✍️` Then, the expression

`\ \ \ \ \ \ \ \ \ \ \ \ \color(red)( a_1 + a_2 + a_3 +,…......+ a_n + .........`

is called the `color(blue)"series"` associated with the given sequence .

`\color{red} ✍️` The series is finite or infinite according as the given sequence is finite or infinite.

`\color{red} ✍️` Series are often represented in compact form, called `color(red)"sigma notation,"` using the Greek letter `color(blue)(Σ "(sigma)")` as means of indicating the summation involved.

`\color{red} ✍️` Thus, the series `color(red)(a_1 + a_2 + a_3 +,…+ a_n)` is abbreviated as `color(blue)(Σ_(k=1)^(n) a_k)`

`\color{red} ✍️` Then, the expression

`\ \ \ \ \ \ \ \ \ \ \ \ \color(red)( a_1 + a_2 + a_3 +,…......+ a_n + .........`

is called the `color(blue)"series"` associated with the given sequence .

`\color{red} ✍️` The series is finite or infinite according as the given sequence is finite or infinite.

`\color{red} ✍️` Series are often represented in compact form, called `color(red)"sigma notation,"` using the Greek letter `color(blue)(Σ "(sigma)")` as means of indicating the summation involved.

`\color{red} ✍️` Thus, the series `color(red)(a_1 + a_2 + a_3 +,…+ a_n)` is abbreviated as `color(blue)(Σ_(k=1)^(n) a_k)`

`color(red)"Key Concept"` When the series is used, it refers to the indicated sum not to the sum itself.

For example, `color(red)(1 + 3 + 5 + 7)` is a finite series with four terms.

When we use the phrase `color(blue)"“sum of a series,”"` we will mean the number that results from adding the terms, the sum of the series is `16.`

For example, `color(red)(1 + 3 + 5 + 7)` is a finite series with four terms.

When we use the phrase `color(blue)"“sum of a series,”"` we will mean the number that results from adding the terms, the sum of the series is `16.`

Q 3009112018

Write the first three terms in each of the following sequences defined by the following:

(i) `a_n = 2n+5`

(ii). `a_n = (n-3)/4`

(i) `a_n = 2n+5`

(ii). `a_n = (n-3)/4`

(i) Here `a_n = 2n + 5`

Substituting `n = 1, 2, 3,` we get `a_1 = 2(1) + 5 = 7, a_2 = 9, a_3 = 11`

Therefore, the required terms are 7, 9 and 11.

(ii). Here `a_n = (n-3)/4` . Thus `a_1 = (1-3)/4 = -1/2 , a_2 = -1/4 , a_3 = 0`

Hence, the first three terms are `-1/2 , -1/4` and `0`

Q 3059212114

What is the 20th term of the sequence defined by `a_n = (n-1)(2-n)(3+n) ?`

Putting `n = 20` , we obtain

`a_(20) = (20 – 1) (2 – 20) (3 + 20)`

`= 19 × (– 18) × (23) = – 7866.`

Q 3019212119

Let the sequence an be defined as follows: `a_1 = 1, a_n = a_(n – 1) + 2` for `n ≥ 2.` Find first five terms and write corresponding series.

We have

`a_1 = 1, a_2 = a_1 + 2 = 1 + 2 = 3, a_3 = a_2 + 2 = 3 + 2 = 5,`

`a_4 = a_3 + 2 = 5 + 2 = 7, a_5 = a_4 + 2 = 7 + 2 = 9.`

Hence, the first five terms of the sequence are 1,3,5,7 and 9. The corresponding series

is `1 + 3 + 5 + 7 + 9 +...`