Consider a, b, c in a G.P. such that |a + b + c| = 15. The median of

Question

Consider a, b, c in a G.P. such that |a + b + c| = 15. The median of these three terms is a, and b = 10. If a > c, what is the product of the first 4 terms of this G.P.?

A. 8,000
B. 16,000
C. 32,000
D. 40,000
E. 48,000

KarishmaB · Accepted Answer

a, b and c are in a GP which means they are a, ar and ar^2 respectively. Median is a and a > c so a < b. In increasing order, it looks like b ,a , c

Now this is odd. GPs usually go in a single direction. If r > 1, then the terms keep increasing and if r lies between 0 and 1, the terms keep reducing. So something must be negative here to get positive-negative terms. We know that b = ar = 10 which is positive, so both a and r must be negative.
Since the sum of three terms is 15, it seems likely that we have all integers.

ar = 10 = -1*-10 = -2*-5 = -5*-2 = -10*-1

|a + ar + ar^2| = 15 
This means that r must be a relatively small value else the sum might shoot up. Try a = -5, r = -2

|-5 + -5*-2 + -5*(-2)^2| = 15 
Valid

So product of first 4 terms =  a*ar*ar^2*ar^3 = (-5)^4 * (-2)^6 = 40,000

Answer (D)

chetan2u · Answer

I believe "Consider a, b, c in a G.P. " means a, b and c ARE in GP . It could easily mean the GP is {x, b, a, c...} and a, b, c are three terms in the GP. Also, the question should specify that a, b, c are first three terms of the GP. Logically The product of the first four terms will be a* ar* ar^2* ar^3=a^4*r^6=(a^2r^3)^2 , that is a SQUARE. Now, |a+b+c|=15 does point towards all the terms to be integer, and at least none of them is a square root or cube root. So our answer should be something that can be written as a SQUARE of something. We cannot have THREE 0s in a square. Only D possible => 40,000 D Algebraically a, b, c are in GP, and b>a>c => ar>a>ar^2 1) ar>a......ar-a>0.....a(r-1)>0 . If a>0 , r>1 OR if a<0, r<1. 2) a>ar^2......a-ar^2>0.....a(1-r^2)>0 . If a>0 , 1>r^2 OR if a<0, r^2>1 . But when a>0, both r>1 and 11. Both r<1 and r^2>1 are possible only when r<-1. Hence a<0 and r<-1 Now ar=10=-2*-5=-5*-2, so a is -2 or -5 But |a+b+c|=15........|a+ar+ar^2|=|a(1+r+r^2)|=15=-3*-5 . so, a has to be a multiple of 3 or 5. Thus, a=-5 and r=-2 We are looking for a*ar*ar^2*ar^3=a^4*r^6=(-5)^4*(-2)^6=4*10^4=40000 D

nick1816 · Answer

a=\frac{10}{k} 
c=10k

10k+10+\frac{10}{k} = 15  or  10k+10+\frac{10}{k} = -15

case 1-   10k+10+\frac{10}{k} = 15

2k+2+\frac{2}{k} =3

2k^2 -k+2=0

We don't have real roots for this equation

case 2-   10k+10+\frac{10}{k} = -15

2k+2+\frac{2}{k} =-3

2k^2 +5k+2=0

(2k+1)(k+2)=0

k= -2 or  -\frac{1}{2}

since a>c, reject k= -\frac{1}{2}

We get a=-5, b=10 and c=-20

Assuming that a is the first term of the sequence

Product of first 4 terms = -5*10*-20*40 = 40000

CEdward · Answer

Hi is GP supposed to mean geometric progression here?

Bunuel · Answer

Yes, G.P. means geometric progression.

Legallyblond · Answer

how is the ascending order b,a,c it should be c,a,b right? can you please explain this solution? KarishmaB please help with this question how to solve this question? if a median and a>c then ascending order would be c,a,b right?

KarishmaB · Answer

Yes, the increasing order will be c, a, b, not b, a, c.

Kinshook · Answer

DisciplinedPrep

Consider a, b, c in a G.P. such that |a + b + c| = 15. The median of these three terms is a, and b = 10. If a > c, what is the product of the first 4 terms of this G.P.?

First term = a = 10/r
Second terms = ar = b = 10
Third term = c = ar^2 = 10r

a > c ; 10/r > 10r; 1/r > r; r - 1/r <0; (r^2-1)/r < 0;
Case 1: r < 0; r^2 -1 > 0; r < -1
Case 2: r> 0; r^2 - 1< 0 ; r<1; 0 < r< 1; Not feasible since median = a;
r < -1

|a + b + c| = |10/r + 10 + 10r| = 15
|1/r + 1 + r| = 1.5

Product of first 4 terms = 10/r *10 * 10r * 10r^2 = 10000*r^2

Only option D 40000 satisfies the conditions and is of the form

To crosscheck, let us take r = -2; since r < -1

r= -2

First term = a = 10/(-2) = -5
Second term =b = 10
Third term =c = 10(-2) = -20
Fourth term =d = 10(-2)^2 = 40

a = -5 > c = -20

|a + b + c| = |-5+10-20| = |-15| = 15

Product of first 4 terms = (-5)*10*(-20)*40 = 40000

IMO D

Consider a, b, c in a G.P. such that |a + b + c| = 15. The median of

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

Consider a, b, c in a G.P. such that |a + b + c| = 15. The median of

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards