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### Theory:

Vinay visited the nearby market, and he observed, the fruits and vegetables were kept in a nice, peculiar manner. He noticed the specific pattern in the way the fruits are arranged. In every row, the number of fruits is increasing by \(1\) sequentially.

Not only in the market, but we can also observe the specific pattern or sequence at many places.

Example:

Go and visit your staircase. There you can see each stair are maintained at a specific height. Thus it is made by following a certain sequence.

The sequences can be found in mathematics as well. Sounds interesting, right?

Let us look at the set of the number with certain patterns.

5, 8, 11, 14, 17, 20...…

Each term in the above set of numbers is increased by 3. So the given set of numbers are continuously increased by 3.

These numbers belong to a category called “Sequences”.

A real-valued sequence is a function defined on the set of natural numbers and taking real values.

Each element in the sequence is called a term of the sequence. The element in the first position is called the first term of the sequence. The element in the second position is called the second term of the sequence and so on.

The general term of the sequence:

If the \(n^t\)\(^h\) term is denoted by \(a_n\), then \(a_1\) is the first term, \(a_2\) is the second term, and so on. A sequence can be written as \(a_1\), \(a_2\), \(a_3\), \(a_4\).

Let's see several examples to understand it better.

Example:

**1**. \(2\), \(4\), \(6\), \(8\)... is a sequence with the general term \(a_n = 2n\). When we put \(n = 1, 2, 3...\), we get \(a_1 = 2\), \(a_2 = 4\), \(a_3 = 6\), \(a_4 = 8\), …

**2**. 1, 3, 5, 7, ... is a sequence with the general term \(a_n = 2n − 1\). When we put \(n = 1, 2, 3....\), we get \(a_1 = 1\), \(a_2 = 3\), \(a_3 = 5\), \(a_4 = 7\), …

**3**. $\frac{1}{3},\frac{1}{5},\frac{1}{7},\frac{1}{9}$, ... is a sequence with general term $\frac{1}{2n+1}$. When we substitute \(n = 1, 2, 3, ...\), we get \(a_1 = \frac{1}{3}\), \(a_2 = \frac{1}{5}\). \(a_3 = \frac{1}{7}\), \(a_4 = \frac{1}{9}\) ...

**Finite sequence**:

When the number of elements in a sequence is finite, it is known as a finite sequence.

Example:

**1.**\(4\), \(7\), \(10\), ... \(22\).

**2.**$\frac{1}{2},\frac{2}{3},\frac{3}{4},\frac{4}{5},...\phantom{\rule{0.147em}{0ex}},\frac{41}{40}$

**Infinite sequence:**

If the number of elements in a sequence is infinite, it is called an infinite sequence.

Example:

**1**. \(2\), \(6\), \(10\), ...

**2**. $\frac{1}{10},\frac{1}{20},\frac{1}{30},\frac{1}{40},$…

Reference:

Image by Peter H from Pixabay