Question

What is an example of arithmetic sequence?

Answer 1

Hint: Now we know that Arithmetic sequences are the sequence in which the difference between two terms is common. Hence to write an arithmetic sequence all we need is one first term and common difference. Now if a is the first term and d is the difference then the sequence is given by $a+d,a+d+d,a+d+d+d,...$ . Hence each term can be written by adding the difference to the previous term. Hence we have the sequence.

Complete step by step solution:
Now to understand arithmetic sequence let us first understand the concept of sequence and series.
Now a sequence is nothing but just an array of things. Hence a sequence is just a list where in each position we have some term. Hence when a sequence is defined for any n which is a natural number we should be able to give its $n^{th}$ term. In general the $n^{th}$ term of a sequence is denoted by $a_n$
Now sequences can be any kind of list and need not necessarily have a pattern. But to study the properties of certain sequences we will work on sequences in pattern. Now the two most general kinds of sequences that we come across are Arithmetic and Geometric sequence.
Now Arithmetic sequence is a sequence in which the difference between two terms is constant and the geometric sequence is a sequence in which the ratio of two terms is constant. Hence for arithmetic sequence if the common difference is d then $a_n-a_{n-1}=d$ and if for a geometric sequence the common ratio is r then we have ${\displaystyle \frac{a_n}{a_{n-1}}}=r$ .
Now let us form an arithmetic sequence.
To form an arithmetic sequence we must first have two quantities the first term and the common difference. Let us say the first term is 0 and the common difference is 1. Then we will form each term by adding 1 to the previous term. Hence we have the sequence,
$\Rightarrow 0,1,2,3,4,5,6,7,8,9...$
Note that here $n^{th}$ term of the sequence is given by $a_n=n-1$ .

Note: Now there is another common sequence that we use which is known as HP. The sequence HP is defined as reciprocals of terms of AP. Hence if we have $1,2,3,4,5,6,7,8,9,..$ are in AP then ${\displaystyle \frac{1}{1}},{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{3}},{\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{5}},....$ are in HP.