The sequence x1, x2, ... xn is such that x1 = 5, x2 = 5, x3 = 0, x4 =

Question

The sequence  x_1 ,  x_2 , ...,  x_n  is such that  x_1 = 5 ,  x_2 = -5 ,  x_3 = 0 ,  x_4 = - 2 ,  x_5 = 4  and  x_n =x_{ n-5} . P is an integer greater than 5. P = ?

(1) Sum of the first P terms in the given sequence is 10.
(2) Sum of the first P + 3 terms in the given sequence is 10.

chetan2u · Answer

Hi..

I am sure the  X n = X n-5   actually means   X_n = X_{n-5} 
 so the sequence is .. 
 5,-5,0,-2,4,5,-5,0,-2,4.........

lets see the statements...
 (1) Sum of the first P terms in the given sequence is 10. 
 lets see how the SUM increases... 
5+(-5)=0....0+0=0.....0+(-2)=-2....-2+4=2.....2+5=7.....7_5=2
so if we are looking at SUM as 10, there should be 5 repetitions as EACH repetition gives us 2...
so 25 numbers in sequence will have 10 as SUM
 But when we add 5 and -5 are added, SUM will again be 10 and then when we add 0(28th number), Sum will again be 10 
so ANS can be 25, 27 or 28
Insuff

(2) Sum of the first P+3 terms in the given sequence is 10. 
 As in statement I, P+3 could be 25,27 or 28 
Insuff

combined..
P can be 25, 27 or 28
 if P is 25, P + 3 =28......POSSIBLE 
if P =27, P+3=30....NO
if P+28, P+3=31...NO

so P = 25
sufficient

C

abhi2707 · Answer

It talks about the "first" P terms...
answer should be D

VeritasPrepBrian · Answer

I like this one. Notice that for every trip through the cycle:

The first two terms net to a sum of zero (5 + -5), so the sum at the end of the cycle is the same as the sum two terms later.

The third term doesn't change the sum, so the sum at the end of the cycle is the same as it is two terms later and three terms later.

You can see this if you list out maybe the first three cycles (so 15 terms) of tracking the sum at each term. You start with the first term of 5, then add -5 to get 0, then add 0 to stay the same, then add -2 to get -2, then add 4 to get 2. Keep doing that and you'll have:

5, 0, 0, -2, 2, 7, 2, 2, 0, 4, 9, 4, 4, 2, 6

Notice how the sets of 2, 7, 2, 2 and 4, 9, 4, 4 include lots of repeat values in them. It's because the cycle is designed that way (on purpose by the testmaker) - if you're keeping track of sums, the 5 then -5 consecutive pair will add 5 then get you right back, then the 0 keeps you where you were before you ever added that 5 in the first place.

Also note that at the end of each cycle you end up having just added 2 to the total (5, -5, 0, -2, 4 nets to +2 when you sum up all five terms).

So for each statement alone, you know that 10 is going to show up multiple times in this sequence. Once you're five cycles in you'll have inched up 2 each cycle to have gotten to 10. But you know its going to repeat - just follow the 2s and 4s in the shorter list above to see how that works. So neither alone can be sufficient...that sum of 10 will show up several times.

*But* when you take the statements together you know you're looking for it to appear three spaces apart at spots P and P-3. That's only going to happen once: with 2 in the list above it happens at the 8th and 5th spots, with 4 it happens at the 13th and 10th spots, etc. It'll be at the end of one cycle of five terms and then three spots later (at terms 28 and 25), and that will be the only increment of three-apart 10s on the list, so you have your answer.

To me a huge key here is seeing that run of 5, -5, 0 - as soon as I see that that indicates to me that something's up...they didn't pick those numbers accidentally, so when I realize that two terms net to a sum of 0 and the third does the same, then I know that if they're asking me about a sum I'm going to want to pay attention there.

And then like so many other sequence problems, even if you don't take the time to get all the way deep into the sequence to the terms (here the 25th and 28th) that they're asking about, if you list out enough of the sequence to find a pattern you can usually see what's going on there and then extrapolate that pattern to the larger values they're asking about.

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