Write an explicit formula for a_na n ​ , the n^{\text{th}}n th term of the sequence 5, -25, 125,

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:

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:

And if we want to write this in a more general way, we can rewrite this as:

Where we removed the necessity of knowing the previous term.