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Sum of Integers (really annoying)

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by GMATGuruNY » Wed Jun 01, 2016 6:23 am
WRITE IT OUT and LOOK FOR A PATTERN.

First term = 1/1 - 1/2 = 1/2.
Second term = 1/2 - 1/3 = 1/6.
Third term = 1/3 - 1/4 = 1/12.
Fourth term = 1/4 - 1/5 = 1/20.

Sum of the first 2 terms:
1/2 + 1/6 = 2/3.

Sum of the first 3 terms:
2/3 + 1/12 = 3/4.

Sum of the first 4 terms:
3/4 + 1/20 = 4/5.

In each case:
The NUMERATOR of the sum = the NUMBER OF TERMS.
THE DENOMINATOR of the sum = NUMERATOR + 1.

Thus:
Sum of the first 100 terms = 100/101.

The correct answer is D.
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by AndreiGMAT » Wed Jun 01, 2016 7:00 am
Mitch, that is surely an interesting approach. Thank you.
But why "traditional" method doesn't work for me here?

My way of thinking:

a1 = 1/2
a100 = 1/10100
Number of integers = 100
Average = 5051/20200
Sum = (5051/20200)*100 = 5051/202

This method always worked with integers for me before.
What am I missing here?
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by GMATGuruNY » Wed Jun 01, 2016 7:05 am
AndreiGMAT wrote:Mitch, that is surely an interesting approach. Thank you.
But why "traditional" method doesn't work for me here?

My way of thinking:

a1 = 1/2
a100 = 1/10100
Number of integers = 100
Average = 5051/20200
Sum = (5051/20200)*100 = 5051/202

This method always worked with integers for me before.
What am I missing here?
The method above is valid for an EVENLY SPACED SET: a set in which the difference between successive terms is always THE SAME.
Here, the first four terms -- 1/2, 1/6, 1/12, 1/20 -- indicate that the set is NOT evenly spaced.
The difference between 1/2 and 1/6 is not the same as the difference between 1/6 and 1/12.
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by Brent@GMATPrepNow » Wed Jun 01, 2016 7:15 am
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
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by GMATGuruNY » Wed Jun 01, 2016 7:16 am
Alternate approach:

a� + a₂ + a₃ = (1/1 - 1/2) + (1/2 - 1/3) + (1/3 - 1/4)
Notice that the portion in red CANCELS OUT, leaving only the FIRST FRACTION AND THE LAST FRACTION.
Thus:
a� + ... + a�₀₀ = (1/1 - 1/2)...+ (1/100 - 1/101).
Since the portion in red will cancel out, we get:
a� + ... + a�₀₀ = 1/1 - 1/101 = 100/101.

The correct answer is D.
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by AndreiGMAT » Wed Jun 01, 2016 7:34 am
Thank you, this is new and extremely useful.
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by [email protected] » Wed Jun 01, 2016 9:05 am
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

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