`\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`.
`\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`.
