Counting Integers

[QuinganZ341]

Could one of the instructors please verify the following understanding for me?

When counting how many integers are in a set of consecutive integers (between 100 and 200, inclusive):
(last - first) +1

When counting how many integers are in a set of consecutive multiple integers:
(last - first)/increment +1

Here, I am not clear on what's a consecutive multiple... Are consecutive multiples the ones that go like 2, 4, 8, 16... (times 2)? Not the the ones that go like 2, 4, 6, 8, 10... (plus two?)

Thank you,
Quing

[RonPurewal]

Could one of the instructors please verify the following understanding for me?

When counting how many integers are in a set of consecutive integers (between 100 and 200, inclusive):
(last - first) +1

this is the kind of thing you should test yourself, using easy/obvious cases.

for instance, it's (hopefully) obvious that, if i list the integers from 1 to 87 (inclusive), the list will contain 87 integers.
here your formula gives 87 – 1 + 1 = 87. so, good.

if you try your formula out on three or four 'easy' cases and it works every time, then you can rest assured that you have a correct formula.

[RonPurewal]

When counting how many integers are in a set of consecutive multiple integers:
(last - first)/increment +1

Here, I am not clear on what's a consecutive multiple... Are consecutive multiples the ones that go like 2, 4, 8, 16... (times 2)? Not the the ones that go like 2, 4, 6, 8, 10... (plus two?)

i don't think i've ever seen the term 'consecutive multiple integers' before.

in this context, though, it's pretty clear that we're talking about the latter case, because the word 'increment' is there.
'increment' refers very specifically to an added value. (if you multiply by something, then it's not an 'increment'.)

the more common name for this sort of thing is 'arithmetic sequence'. (you will NOT be responsible for any such terminology on the gmat exam.)

[RonPurewal]

as long as we're on the topic, i may as well mention how i personally handle this sort of thing.

see, i am pretty much constitutionally incapable of remembering formulas. (if you've spent much time on this forum, you've probably seen many paeans to my horrible memory.)
so, instead, i just manipulate the sequence with arithmetic until it has the form '1, 2, 3, ..., n'. and then, of course, there are exactly n integers.

example:
17, 22, 27, ..., 502

here the 'increment' is five. so, first subtract 12, to transform into consecutive multiples of five:
5, 10, 15, ..., 490

now divide by five:
1, 2, 3, ..., 98

okay, there are 98 integers in that list.

[QuinganZ341]

i just manipulate the sequence with arithmetic until it has the form '1, 2, 3, ..., n'. and then, of course, there are exactly n integers.

example:
17, 22, 27, ..., 502

here the 'increment' is five. so, first subtract 12, to transform into consecutive multiples of five:
5, 10, 15, ..., 490

now divide by five:
1, 2, 3, ..., 98

okay, there are 98 integers in that list.

THANK YOU! This really helps.. and ironically, it's helping me with memorizing the formula a lot more because now I actually understand it...

[tim]

Bingo! That's EXACTLY how it should work. If you understand something, distilling it down to something you can memorize is easy. But if you just try to memorize something without understanding it, you will have a lot of difficulty.

[RonPurewal]

hmm-- ironically, that reasoning doesn't help me understand the formula any better at all. (rather, it just obviates the need for the formula in the first place.)

--

if you asked me for something to help you understand the formula, i'd give you this:
here are some sticks, spaced at intervals of (let's say) 1 mile.
|__________|__________|__________|__________|__________|__________|
you're trying to count the sticks.

on the other hand, subtracting numbers gives you a DISTANCE between sticks (NOT a count number).

when the COUNT number is '1', the DISTANCE is zero. (the distance between the first stick and itself is ... zero.)
from this point onward, each additional stick adds 1 mile to the distance, so the count number will be exactly 1 more than the distance, forever and ever and ever.

(clearly it's hard to put this into words, but i think you can tell what i mean.)