Given the following 2 arithmetic sequences, determine the first ten terms common to both sequences.
3, 14, 25, ...
2, 9, 16, ...
I figured out that the two general formulas would be tn = 3 + 11 (n-1) and tn = 2 + 7 (n-1), respectively... But I'm not sure where to go from there. I've asked for help elsewhere but I still don't get it, and my test is coming up soon... =/ Thanks a lot.
If I'm reading your original post correctly, you don't want the first ten terms of each sequence, right? You want the first ten terms that are COMMON to BOTH sequences. If so:
Here's what I did - it's slightly better than listing out every term.
The first sequence's terms can be described by 2 + 7x.
The second sequence's terms can be described by 3 + 11y.
I want to know when these are equal, so:
2 + 7x = 3 + 11y.
I solved for y and got:
So I'm looking for an integer value of x that gives me an integer value for 7 (since all the numbers in both sequences are integers). I found the first one by trial and error. x=8 gives y=5. So the point (8,5) is on my line, and gives 58 for both 2 + 7x and 3 + 11y. (Points in which both coordinates are integers are called "lattice points" - there may be an easier way to find them, but I don't know it. Essentially, that's what we're looking for here - lattice points on our line.)
So 58 is in both sequences. Now, noting that the slope of our line is 7/11, we can increase our y value by 7 and our x value by 11 (since slope is change in y / change in x) to get the next value of x and y that gives integer coordinates: (19, 12). Plugging x = 19 and y = 12 into the appropriate equations gives 135 in both, so that's the next term.
Keep using slope to "jump" from each integer-coordinate point to the next. Use x and y, plug them into your equations, and you should be good to go:
(8,5), (19,12), (30, 19), (41, 26), ...
Hope this makes sense. I kind of figured it out while writing this, so it's a little... disorganized.