Question

Find an explicit form f(n) for each of the following arithmetic sequences (assume a is some real number and x is

some real number).
b. 1 / 5, 1 / 10, 0, − 1 / 10, ...

Answer 1

The explicit form for this sequence is
`f(n)=f(1)-(n-1)/(10)` for all
`n\geq 1`.

Explanation:

The explicit form of an arithmetic sequence of numbers is given by the formula
`f(n)=f(1)+(n-1)d`, where
`f(1)` is the first term of the sequence,
`d` is the difference between two consecutive terms of the sequence, and
`n\geq 1`.

We know that the first four elements for the arithmetic sequence are
`{f(1)=(1)/(5),f(2)=(1)/(10),f(3)=0, f(4)=-(1)/(10)}`.

To find the general formula for this problem we only need to calculate
`d` in the above formula.

For n=2, we have

`f(2)=f(1)+(2-1)d=f(1)+d`

if we replace
`f(1)=(1)/(5)` and
`f(2)=(1)/(10)` and solve for
`d` we obtain

`(1)/(10)=(1)/(5)+d`

`d=(1)/(10)-(1)/(5)=-(1)/(10)`

Therefore the explicit form is
`f(n)=f(1)-(n-1)/(10)` for all
`n\geq 1`.