I'm having some difficulty understanding finding patterns to the nth degree.
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.
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
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.
To write the pattern as a final answer, you can first observe the relationship between the numerators and denominators:
Denominators: 1, 3, 9, 27, 81, 243
The denominators follow a pattern where each term is obtained by multiplying the previous term by 3.
Numerators: -1, 1/3, 7/9, 25/27, 79/81, 241/243
The numerators follow a pattern of being two less than the corresponding denominator.
Based on these observations, you can express the pattern as follows:
To find the nth term, the numerator will be (3^n - 2) and the denominator will be 3^n.
For example, to find the value of the 7th term:
Numerator: (3^7 - 2) = 2186 - 2 = 2184
Denominator: 3^7 = 2187
Therefore, the 7th term of the pattern is 2184/2187.
In general, you can write the nth term of the pattern as follows:
Numerator = (3^n - 2)
Denominator = 3^n
I hope this helps! Let me know if you have any further questions.