Answer:
[tex]a_n = (-1)^{n + 1}*(5)^n[/tex]
Step-by-step explanation:
First, let's analyze the sequence.
We start with a 5.
Then, we get -25
We know that 25 is 5^2, and there is also a change of sign.
Then, we get 125.
We again know that 125 = 25^2, and again we have a change of sign.
So we will have something like:
[tex]a_n = (-1)^{n + 1}*a_{n-1}^2[/tex]
Where the (-1)^n part is the responsible for the change of sign.
If we have:
a₁ = 5
Then:
a₂ = (-1)^3*(a₁)^2 = -1*(5)^2 = -25
as expected.
and:
a₃ = (-1)^4*(a₂)^2 = 1*25^2 = 125
as expected.
Then we can conclude that the explicit formula for the Nth therm is:
[tex]a_n = (-1)^{n + 1}*a_{n-1}^2[/tex]
And if we want to write this in a more general way, we can rewrite this as:
[tex]a_n = (-1)^{n + 1}*(5)^n[/tex]
Where we removed the necessity of knowing the previous term.
