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:

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

Explanation:

Given that:

$\begin{gathered} a_1=75 \\ a_n=-\frac{1}{5}a_{n-1} \end{gathered}$

We have the following:

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

The sequence is:

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

This is alternating, so,

$a_n=(-1)^{n+1}.\frac{75}{5^n}$

Where

$n\ge1$