If I have an equation and an inequality - normally I would insert the equation into the inequality. Like this for example:
However, take a look at this problem here:
Some sequence is such that Sn = 1/n - 1/(n+1) for all integers n=>1. If k is a positive integer, is the sum of the first k terms of the sequence greater than 9/10?
Here is how I solved this:
1) I took some primes and played around with Sn = 1/n - 1/(n+1) to figure out that the sum of any consecutive integers will be 1/first less 1/last.
2) I took a look at the first statement, k>10. Ok let's try k=10 this gave me 9/10. The question asks if the sum is greater than 9/10, so I must figure out which way the inequality points. Because I didn't know how to do it, I simply took the next number, i.e. 11 and got the answer 10/11, which is bigger, so YES. Also this means SUFFICIENT.
3) Statement 2 obviously was NOT suff, since it could go all the way to 0 and from my above experimentation I already figured out which way the inequality points.
I have seen MGMAT write down fractions in this form: something over greater than something. So for example 2/>4. So I was wondering - is there anyway I could have build a similar equation with 10 when I was trying to figure out which way the sign of the inequality was pointing? In other words, how could I have shortened / eliminated the need to do both 10 and then 11 to figure out where the inequality points?
Am I completely offtrack here? Is there a much quicker solution I am missing?
Took me 2:44
First note that this questions is not a GMAT style question. You cannot have two different ranges of k in the two statements with no overlap.
This is how you deal with sequences: http://www.veritasprep.com/blog/2012/03 ... sequences/
Write down the first few terms:
S1 = 1 - 1/2
S2 = 1/2 - 1/3
S3 = 1/3 - 1/4
and so on...
Note that the second fraction of every term is same as first fraction of subsequent term. So when you add some terms, you are left with only the first and the last fraction. e.g.
S1 + S2 + S3 = 1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4
S1 + S2 + S3 = 1 - 1/4
Similarly, S1 + S2 + S3 + S4 + S5 + S6 = 1 - 1/7
To get 9/10, we need 1 - 1/10 which means it is the sum of S1 + S2 + S3 + ... S7 + S8 + S9
If you instead find the sum of the first 10 terms, S1 + S2 + ... + S9 + S10 = 1 - 1/11 (Note that we are subtracting a number less than 1/10 from 1 here. This means the sum will be more than 9/10).
So you know that when you sum the first 9 terms, you get 9/10. If you add more terms, you will get a sum which is more than 9/10. If you add fewer terms, you will get a sum which is less than 9/10.
Also writing fractions with inequalities is an informal way of writing. 2/(>4) means the denominator is greater than 4 and to save words, we write it like this. We cannot do any calculations like this.
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