Answer:
The explicit form for this sequence is [tex]f(n)=f(1)-\frac{n-1}{10}[/tex] for all [tex]n\geq 1[/tex].
Step-by-step explanation:
The explicit form of an arithmetic sequence of numbers is given by the formula [tex]f(n)=f(1)+(n-1)d [/tex], where [tex]f(1) [/tex] is the first term of the sequence, [tex]d[/tex] is the difference between two consecutive terms of the sequence, and [tex]n\geq 1[/tex].
We know that the first four elements for the arithmetic sequence are [tex]{f(1)=\frac{1}{5},f(2)=\frac{1}{10},f(3)=0, f(4)=-\frac{1}{10}}[/tex].
To find the general formula for this problem we only need to calculate [tex]d[/tex] in the above formula.
For n=2, we have
[tex]f(2)=f(1)+(2-1)d=f(1)+d [/tex]
if we replace [tex]f(1)=\frac{1}{5}[/tex] and [tex]f(2)=\frac{1}{10}[/tex] and solve for [tex]d[/tex] we obtain
[tex]\frac{1}{10}=\frac{1}{5}+d[/tex]
[tex]d=\frac{1}{10}-\frac{1}{5}=-\frac{1}{10}[/tex]
Therefore the explicit form is [tex]f(n)=f(1)-\frac{n-1}{10}[/tex] for all [tex]n\geq 1[/tex].
