SOLUTION: A sequence begins with 1, 4, 7, … Another sequence begins with 99, 95, 91, … At which position is the positive difference between the respective terms of the two sequences the mi

Algebra ->  Sequences-and-series -> SOLUTION: A sequence begins with 1, 4, 7, … Another sequence begins with 99, 95, 91, … At which position is the positive difference between the respective terms of the two sequences the mi      Log On


   

Question 260663: A sequence begins with 1, 4, 7, … Another sequence begins with
99, 95, 91, … At which position is the positive difference between
the respective terms of the two sequences the minimum?
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
Both of these sequences are arithmetic sequences. The general formula for the terms of an arithmetic sequence is:
a%5Bn%5D+=+a%5B1%5D+%2B+%28n-1%29d
For the first sequence, since the first term, a%5B1%5D, is 1 and the common difference, d, is 3 we get:
a%5Bn%5D+=+1+%2B+%28n-1%293
For the second sequence we get:
a%5Bn%5D+=+99+%2B+%28n-1%29%28-4%29
The positive difference in these terms, which we'll call D, is:
D+=+abs%28%281+%2B+%28n-1%293%29+-+%2899+%2B+%28n-1%29%28-4%29%29%29
(Note how we use absolute value to ensure a positive difference.) Simplifying this we get:
D+=+abs%28%281+%2B+3n-3%29+-+%2899+%2B+-4n+%2B+4%29%29
D+=+abs%28%283n-2%29+-+%28103+%2B+-4n%29%29
D+=+abs%287n+-+105%29
So the question is: What value of n makes 7n - 105 closest to zero? After a little effort we find that if n = 15, then 7n - 105 is zero! In other words, the 15th term of both sequences is the same!