Arithmetic Sequence? Find the first five terms of the sequence,

To find: 
The sequence's first five terms to determine whether or not it is an arithmetic sequence. If it is arithmetic sequence, find the common difference, and express the nth term of the sequence in the standard form ${\displaystyle a_n=a+(n-1)d}$ 
Concept used: 
In an arithmetic sequence, there is a fixed difference between each succeeding term.
An arithmetic sequence's nth term is determined by,
${\displaystyle a_n=a+(n-1)d}$ .......(1) 
Calculation: 
Given sequence is, 
${\displaystyle a_n=1+\frac{n}{2}}$ .......(2) 
Substitute 1 for n in equation (2) to calculate the first term of this sequence. 
${\displaystyle a_1=1+\frac{1}{2}}$ 
${\displaystyle =\frac{3}{2}}$ 
Substitute 2 for n in equation (2) to calculate the second term of this sequence. 
${\displaystyle a_2=1+\frac{2}{2}}$ 
${\displaystyle =1+1}$ 
${\displaystyle =2}$ 
Substitute 3 for n equation (2) to calculate the third term of this sequence. 
${\displaystyle a_3=1+\frac{3}{2}}$ 
${\displaystyle =\frac{5}{2}}$ 
Substitute 4 for n equation (2) to calculate the fourth term of this sequence. 
${\displaystyle a_4=1+\frac{4}{2}}$ 
${\displaystyle =1+2}$ 
${\displaystyle =3}$ 
Substitute 5 for n equation (2) to calculate the fifth term of this sequence. 
${\displaystyle a_5=1+\frac{5}{2}}$ 
${\displaystyle =\frac{7}{2}}$ 
Difference between first and second term can be calculated as, 
${\displaystyle a_2-a_1=2-\frac{3}{2}}$ 
${\displaystyle =\frac{1}{2}}$ 
Difference between second and third term can be calculated as, 
${\displaystyle a_3-a_2=\frac{5}{2}-2}$ 
${\displaystyle =\frac{1}{2}}$ 
Difference between third and fourth term can be calculated as, 
${\displaystyle a_4-a_3=3-\frac{5}{2}}$ 
${\displaystyle =\frac{1}{2}}$ 
Difference between fourth and fifth term can be calculated as, 
${\displaystyle a_5-a_4=\frac{7}{2}-3}$ 
${\displaystyle =\frac{1}{2}}$ 
Since, the difference between the successive terms of the given sequence is constant. 
Therefore, the sequence ${\displaystyle a_n=1+\frac{n}{2}}$ is not arithmetic sequence. 
First term of the sequence ${\displaystyle a_n=1+\frac{n}{2}}$ is ${\displaystyle a=\frac{3}{2}}$. 
Common difference of the sequence ${\displaystyle a_n=1+\frac{n}{2}}$ can be calculated as, 
${\displaystyle d=a_2-a_1}$ 
${\displaystyle =\frac{1}{2}}$ 
Substitute ${\displaystyle \frac{3}{2}}$ for a and ${\displaystyle \frac{1}{2}}$ for d in equation (1) to obtain the expression for ${\displaystyle a_n}$. 
${\displaystyle a_n=a+(n-1)d}$ 
${\displaystyle =\frac{3}{2}+(n-1)\frac{1}{2}}$ 
Conclusion: 
Hence, the sequence ${\displaystyle a_n=1+\frac{n}{2}}$ is an arithmetic sequence, the common difference d is ${\displaystyle \frac{1}{2}}$ and the nth term of the sequence in e standard form is