12 Days of Christmas GMAT Competition - Day 7: j, 9, k, and m are in a

Question

12 Days of Christmas GMAT Competition with Lots of Fun

j, 9, k, and m are in arithmetic progression (in the given order) and j, k, and m are in geometric progression (in the given order). What is the value of j*k? (An arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant. A geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.)

(1) The sum of k+3 and 2m is divisible by 5
(2) The sum of j and 9 is greater than the sum of m and k

Bunuel · Accepted Answer

GMATWhiz Official Explanation:

Step 1: Analyse Question Stem:  
    Since j, 9, k and m are in AP, we can say: 
 o j = 9 – d
o 9
o k = 9 + d
o m = 9 + 2d 
  Also, since j, k and m are in GP, we can say: 
 o (9-d)(9+2d) = (9+d)^2
⇒81 + 9d – 2d^2= 81 + d^2+ 18d 
⇒3d^2+ 9d = 0 
⇒d^2+ 3d = 0   
  So, d = 0 or d = - 3

We need to find the value of jk = (9-d)(9+d)=81-d^2

Step 2: Analyse Statements Independently (And eliminate options) – AD/BCE

Statement 1:  The sum of k+3 and 2m is divisible by 5 
   According to this statement, k + 3 + 2m = 5k where k is a positive integer 
 ⇒9+d+3+2(9+2d)=5k 
⇒12+d+18+4d=5k 
⇒30+5d=5k   
  The above equation is satisfied by both d = 0 and d = -3  
  So, we do not know the exact value of d and thus the exact value of 81-d^2

Thus, statement 1 alone is not sufficient and we can eliminate options A and D

Statement 2:  The sum of j and 9 is greater than the sum of m and k 
   According to this statement, 18 – d > 18 + 3d
 ⇒4d < 0 
⇒d < 0   
  Between d = 0 and d = -3, only d = -3 will satisfy d < 0 and so the value of d is -3

Hence the right answer is Option B.

Surerere · Answer

j,9,k,m are in arthmatic progress so j+k=18
The possible values of j,k is the group of (1,17),(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10) and (9,9) he

So the possibility of series is here
1,9,17,25
17,9,1,-7
2,9,16,23
16,9,2,-5
3,9,15,21
15,9,3,-3
4,9,14,19
14,9,4,-1
5,9,13,17
13,9,5,1
6,9,12,15
12,9,6,3
7,9,11,13
11,9,7,5
8,9,10,11
10,9,8,7
9,9,9,9
Then in the question given thet j,k,m are in geometric pattern
So here are only two possibilities
12,9,6,3
And 
9,9,9,9
Now applied statement 1st so we can see it's satisfied both series so it's not able to provide solutions

Now considered 2nd statement then we find its can discrimination in both statements and provide answer of question.

So Ans- B

Posted from my mobile device

Lizaza · Answer

Let's visualise the two sequences:
 j  (+d)  9  (+d)  k  (+d)  m 
 j  (*a)  k  (*a)  m

Interestingly, the difference between k and m is just once  d , but between j and k - it's twice as big, it's  d  two times.
This means that in the geometric progression is not growing - it seems to be diminishing. So, the ratio of the progression is definitely not higher than 1. 
Let's calculate it:
 k - j = 2d
m - k = d 
Therefore,  k - j = 2(m - k)
aj - j = 2(ak - k)
j(a-1) = 2k(a-1)
j(a-1) - 2k(a-1) = 0
(a-1)(j-2k) = 0 
From here, we get two options for the common ratio:  a = 1  and  a = 0.5 
I know that theoretically, it looks odd if all the members of the progression are equal - but we have nothing stating that this is impossible.
Let's finalise these two scenarios and then look at the given conditions:
   a=1 , so  j=k=m=9  
  a=0.5 , so  k = \frac{j}{2} , which means  2d = \frac{j}{2}   and  d=\frac{j}{4} , 
so  j = 9 + \frac{j}{4}  and   j = \frac{9}{3/4} = 12 
And thus,  j = 12, k = 6, m =3

Now let's consider the conditions:
 
 1.  k+3 + 2m = 12 + 18 = 30  : divisible by 5. 
2.  k+3 + 2m = 9 + 6 = 15  : divisible by 5.  
So,   Condition 1 is not enough  , it doesn't help us at all.

Without further talk, this simply tells us that all members of the sequence cannot be equal. This only leaves us one scenario:  j*k=12*6=72 
So,  Condition 2 is enough by itself -  Answer B.

bumpbot · Answer

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