Sum of Integers (really annoying)

term(n) = 1/x - 1/(x+1)

What is the sum of the first 100 terms of this sequence?

A) 0
B) 1/101
C) 99/100
D) 100/101
E) 1

Another approach:
term1 = 1/1 - 1/2
term2 = 1/2 - 1/3
term3 = 1/3 - 1/4
term4 = 1/4 - 1/5
.
.
.
term99 = 1/99 - 1/100
term100 = 1/100 - 1/101

So, the sum looks like this:
Sum = (1/1 - 1/2) + (1/2 - 1/3) + (1/3 - 1/4) + (1/4 - 1/5) + .... + (1/99 - 1/100) + (1/100 - 1/101)
NOTICE THAT MOST TERMS CANCEL OUT
Sum = (1/1 - 1/2) + (1/2 - 1/3) + (1/3 - 1/4) + (1/4 - 1/5) + .... + (1/99 - 1/100) + (1/100 - 1/101)
= 1/1 - 1/101
= 101/101 - 1/101
= [spoiler]100/101[/spoiler]
= D

Cheers,
Brent

Hi AndreiGMAT,

While this question looks complex, there's a great pattern-matching shortcut built into it - to recognize the pattern, you'll likely have to do a bit of work on the pad though.

We're asked for the sum of the first 100 terms in the given sequence, but the GMAT doesn't really expect you to add up all of that work by hand. I'm going to write down the first few terms as a reference, but I'm NOT going to do any of the math yet....

N=1 --> 1/1 - 1/2
N=2 --> 1/2 - 1/3
N=3 --> 1/3 - 1/4
N=4 --> 1/4 - 1/5
Etc.

Notice that when we subtract a fraction, we end up adding it right back in the next calculation (re: -1/2....+1/2....-1/3....+1/3), so a lot of that math 'cancels out.' The only terms that WON'T cancel out are 1/1 and the very last term.... -1/101

Thus, the sum of those first 100 terms ends up totaling 1/1 - 1/101 = 100/101

Final Answer: D

GMAT assassins aren't born, they're made,
Rich