How would I find the expression of the nth term sequence given the terms of 2, 1, 4/5, 5/7, 6/9 and the expression of the nth term sequence given the terms of 1/2,-1/4,1/8, -1/16?
Change the first sequence to ...
2/1; 3/3;4/5; 5/7;6/9;....
On the second, it sure looks like r is -1/2.
To find the expression for the nth term sequence, you need to look for a pattern or relationship between the given terms. Let's work through each sequence of terms individually:
1. Sequence with terms 2, 1, 4/5, 5/7, 6/9:
To find the pattern, let's look at the differences between consecutive terms:
2 - 1 = 1
1 - 4/5 = 1/5
4/5 - 5/7 = 3/35
5/7 - 6/9 = 23/63
From the differences, we can see that the numerator is increasing by 2 each time, and the denominator is increasing by 8 each time.
Hence, the expression for the nth term sequence is:
Numerator: Start with 2 and increase by 2(n-1)
Denominator: Start with 1 and increase by 8(n-1)
So, the expression for the nth term sequence is given by:
Term(n) = [2 + 2(n-1)] / [1 + 8(n-1)]
2. Sequence with terms 1/2, -1/4, 1/8, -1/16:
Again, let's examine the differences:
1/2 - (-1/4) = 3/4
-1/4 - 1/8 = -3/8
1/8 - (-1/16) = 3/16
Here, we can see that the numerator alternates between 3 and -3, and the denominator is consistently a power of 2.
Hence, the expression for the nth term sequence is:
Numerator: Start with 3 and alternate the sign (-1)^(n+1) * 3
Denominator: Start with 2 and keep it constant
Thus, the expression for the nth term sequence is given by:
Term(n) = (-1)^(n+1) * 3 / 2
By following these steps and looking for patterns or relationships among the terms, you can find the expression for any nth term sequence.