I'm having some difficulty understanding finding patterns to the nth degree.

Question

Here is my problem:

-1, 1/3, 7/9, 25/27, 79/81, 241/243

I recognize the pattern in the denominators is multiplying by 3 and the pattern in the numerators is that they are two less than the numerators.

What I don't understand is how to write that as a final answer.

Thank you.

Answer 1

just write that in symbols:

nth denominator is 3^(n-1)
nth numerator is 3^(n-1) - 2

So, the ntheterm is

Tn = (3^(n-1) - 2)/3^(n-1)

That can be massaged in various ways:

Tn = 1 - 2/3^(n-1)
Tn = 1 - 6/3^n
Tn = (3^n - 6)/3^n

Answer 2

Thank you but I guess that's what I'm not understanding. What does the n-1 mean exactly?

And then why is the numerator that with minus 2? In know the numerator is 2 less than the denominator but I guess I'm just not catching on to why you call it that.

Thank you.