Question

how many integers are strickly between the 257th positive odd integer and the 360th positive even integer?

Answer 1

ALWAYS break a problem down and deal with smaller numbers. Experiment. What will 2n get you when 'n' is any given value? Always an even number. How about 2n-1, when 'n' is any given value? Always an odd number. What if we're looking for the 3rd positive odd integer, and not the 257th? How would we find the 3rd? Well, we know that the 3rd is 5. We could figure that out manually. But we know that 2n-1=5, when 'n' is the 3rd positive odd integer. 2(3)-1=5. Perfect. Let's try the 4th. What's the 4th positive odd integer? We know that the 4th is 7, and we could figure that out manually, too. But still, we know that 2n-1=7, when 'n' is the 4th positive odd integer. 2(4)-1=7. Awesome! We have a formula, and we can use it to determine the 257th positive odd integer. 2(257)-1=513, and this must be the 257th positive odd integer. To determine the 360th positive even integer, we use the same approach. How do we determine the 3rd positive even integer? We know that the 3rd is 6. 2n will get you any even integer you want. ( 2(1)=2, 2(2)=4, etc.) 2n=6 for now, and so n = the 3rd positive even integer. Good!!!! Let's try that for the fifth even integer. We know that the 5th is 10. 2n=10, and 'n' turns out to be the 5th positive even integer. So this formula works. Let's take a look at the real question. We're looking for the 360th positive even integer. So, 2n = 2(360) = 720, and 720 is the 360th positive even integer. All we have to do now to find the integers between 720 and 513 is subtract. So Imma do dat.

720-513 = 207! 207 is da number of plain old integers between the 257th positive odd integer and the 360th positive even integer.

Good luck! ^_^