Question

What is the ${{n}^{th}}$ term of a sequence?

Answer 1

For solving these types of questions, firstly we need to understand the sequence and its type i.e. arithmetic and geometric sequence. You can understand it with the help of definitions, formulas, examples and after understanding these all things you will get the required answer.

Complete step by step answer:
A sequence can be defined as an arrangement of elements or numbers in some definite order. We can represent sequence as: ${{x}_{1}},{{x}_{2}},{{x}_{3}},............{{x}_{n}}$ where $1,2,3$ represents the position of every number and here $n$ represents the ${{n}^{th}}$ term.
There are basically two types of sequences- Arithmetic sequence and geometric sequence.
Arithmetic Sequence can be defined as a list of elements or numbers with a definite pattern. In this sequence if we take any number and then subtract it by the previous number, then the difference that will come is always the same or constant. And the constant difference in all the pairs of successive numbers in a sequence is known as the common difference and it is denoted by the letter $d$.
If the common difference is positive, then the sequence is increasing, but when the difference is negative, then the sequence is decreasing.
Geometric Sequence can be defined as the collection of numbers in which each number is a constant multiple of the previous term. In this sequence if we take any number and then divide it by the previous number, then the ratio that we will get is always constant. In this sequence there is a common ratio between the successive terms and this common ratio is denoted by $r$.
In sequence, the ${{n}^{th}}$ term formula is used to find the general term of any given sequence.
The ${{n}^{th}}$ term of an arithmetic sequence is given as:
${{a}_{n}}=a+(n-1)d$
Where, ${{a}_{n}}$represents the ${{n}^{th}}$ term of the sequence
$a$ is the first term of the sequence.
$d$ is the common difference.
The ${{n}^{th}}$ term of a geometric sequence is given as:
${{t}_{n}}=a{{r}^{n-1}}$
Where, ${{t}_{n}}$ represents the ${{n}^{th}}$ term of the sequence
$a$ is the first term.
$r$ is the common ratio.

Note: The Fibonacci sequence was introduced by Italian mathematician known as Fibonacci. This sequence is a set of numbers starting with $0$ and $1$ , then adding the sum of the last two numbers and it is the next number of the sequence. For ex: $0,1,1,2,3,5,8,13,.............$