Common difference = [tex]\frac{1}{5}[/tex]
Solution:
Given arithmetic sequence:
[tex]$-2,-1\frac{4}{5} ,-1\frac{3}{5} , -1\frac{2}{5} , ....[/tex]
Let us first convert the improper fraction into mixed fraction.
[tex]$-2,-\frac{9}{5} ,-\frac{8}{5} , -\frac{7}{5} , ....[/tex]
Difference between two numbers in an arithmetic sequence is the common difference.
[tex]$a_2-a_1=-\frac{9}{5}-(-2)=-\frac{9}{5}+2=\frac{1}{5}[/tex]
[tex]$a_3-a_2=-\frac{8}{5}-\left(-\frac{9}{5}\right)=-\frac{8}{5}+\frac{9}{5}=\frac{1}{5}[/tex]
[tex]$a_4-a_3=-\frac{7}{5}-\left(-\frac{8}{5}\right)=-\frac{7}{5}+\frac{8}{5}=\frac{1}{5}[/tex]
Common difference = [tex]\frac{1}{5}[/tex].
Hence the common difference of the arithmetic sequence is [tex]\frac{1}{5}[/tex].
Common difference of the given arithmetic sequence is equal to [tex]\frac{1}{5}[/tex]
The given arithmetic sequence is in mixed numbers. Converting them into proper fractions, we get:
[tex]First\ term = t_1 = -2\\\\Second\ term = t_2 = -1\frac{4}{5} = -\frac{(5\times1)+4}{5} = -\frac{9}{5}\\\\Third\ term = t_3 = -1\frac{3}{5} = -\frac{(5\times1)+3}{5} = -\frac{8}{5}\\\\Fourth\ term=t_4 = -1\frac{2}{5} = -\frac{(5\times1)+2}{5} = -\frac{7}{5}[/tex]
The sequence can thus be rewritten as [tex]-2, -\frac{9}{5}, -\frac{8}{5}, -\frac{7}{5}[/tex]
To find the common difference of the given arithmetic sequence, subtract any two consecutive numbers of the sequence. The difference between two consecutive numbers is always a constant and is termed as the common difference.
Hence,
[tex]d_1= t_2-t_1=(-\frac{9}{5} -(-2)= \frac{-9+10}{5}= \frac{1}{5} \\\\d_2=t_3-t_2= (-\frac{8}{5} -(-\frac{9}{5} )= \frac{-8+9}{5} = \frac{1}{5}\\\\d_3=t_4-t_3= (-\frac{7}{5} -(-\frac{8}{5} )= \frac{-7+8}{5} = \frac{1}{5}[/tex]
[tex]d_1=d_2=d_3 = d[/tex]
Common difference [tex]d=\frac{1}{5}[/tex]