Question

Write an explicit formula that represents the sequence defined by the followingrecursive formula:a1 =75 and an =an-1

Answer 1

Answer:

The formula for the sequence is:

`\begin{gathered} a_n=(-1)^(n+1).(75)/(5^n) \\ for \\ n\ge1 \end{gathered}`

Step-by-step explanation:

Given that:

`\begin{gathered} a_1=75 \\ a_n=-(1)/(5)a_(n-1) \end{gathered}`

We have the following:

`\begin{gathered} a_2=-(1)/(5)a_1 \\ \\ =-(1)/(5)(75) \\ \\ =-15 \end{gathered}`
`\begin{gathered} a_3=-(1)/(5)a_2 \\ \\ =-(1)/(5)(-15) \\ \\ =3 \end{gathered}`
`\begin{gathered} a_4=-(1)/(5)a_3 \\ \\ =-(1)/(5)(3) \\ \\ =-(3)/(5) \end{gathered}`

The sequence is:

75, -15, 3, -3/5, ...

This is alternating, so,

`a_n=(-1)^(n+1).(75)/(5^n)`

Where

`n\ge1`