Question

given the arithemtic or geometric sequence, identify f(0) and the common difference ratio f(x) = 4-2/3x

Answer 1

Required value of f(0) = 4 and common difference of given sequence is
`(-2)/(3)`

Solution:

Given sequence rule for arithmetic sequence is
`\mathrm{f}(\mathrm{x})=4-\left((2)/(3)\right) x`

We need to determine f(0) and common difference

Calculating f(0).

Substituting x = 0 in given sequence rule we get

`\begin{array}{l}{f(0)=4-\left((2)/(3)\right) 0} \\ {=4-0=4}\end{array}`

Calculating common difference

Let’s first calculate f(1) and f(2).

Substituting x = 1 in given sequence rule

`\begin{array}{l}{\mathrm{f}(1)=4-\left((2)/(3)\right) 1} \\ {=4-\left((2)/(3)\right)=(10)/(3)}\end{array}`

`\mathrm{f}(2)=4-\left((2)/(3)\right) 2=4-\left((4)/(3)\right)=(8)/(3)`

Common Difference can be found by subtracting f(1) from f(2)

`\text { Common difference }=\mathrm{f}(2)-\mathrm{f}(1)=\left((8)/(3)\right)-\left((10)/(3)\right)=(-2)/(3)`

Hence required value of f(0) = 4 and common difference of given sequence is
`(-2)/(3)`