x, 17, (3x - y^2 - 2) and (3x + y^2 - 30) are four consecutive terms

Question

x, 17, (3x - y^2 - 2) and (3x + y^2 - 30) are four consecutive terms of an increasing arithmetic sequence. The sum of the four numbers is divisible by:

A. 2
B. 3
C. 5
D. 7
E. 11

CareerGeek · Accepted Answer

Any 4 consecutive terms of an arithmetic series will always be of the form {a, a+d, a+2d, a+3d}

Sum = a + (a+d) + (a+2d) + (a+3d) = 4a + 6d
—> Sum = 2(2a+3d) which is always divisible by 2

IMO Option A

Pls Hit kudos if you like the solution

Posted from my mobile device

Mohammadmo · Answer

x+17+3x-y²-2+3x+y²-30=7x+15=
7(x+2)+1
 -----
 ⁿ
7n+1 is a number which is surely divisible by 2.
Option A

Posted from my mobile device

chetan2u · Answer

Not necessary..
If n is even, then NO...If n is odd, then yes

ShankSouljaBoi · Answer

Sum of AP = n/2(a+l)
So sum of 4 terms is 4/2(something) = 2*something
Hence , A

Posted from my mobile device

chetan2u · Answer

Ofcourse, I take it as x and y are integers...

Now 17 is the second term and odd....
A) If the common difference between them is ODD, 
2nd and 4th terms will be ODD as 4th=17+Odd+odd=Odd
1st and 3rd terms will be EVEN as 17-Odd and 17+odd will be EVEN
SUM = E+O+E+O=E, so divisible by 2

B) If the common difference between them is EVEN, 
all terms will be ODD as 17-Even, 17+even, 17+even+even will all be odd
SUM = O+O+O+O=E, so again divisible by 2

SO A..

Also 3rd and 4th term tell us that the difference is EVEN
Ofcourse as mentioned above, the terms will be 17-a, 17, 17+a, 17+2a.. Sum =17*4+(21+a-a)=2(17*2+a).. Hence divisible by 2

chetan2u · Answer

Let the four terms x, 17, (3x - y^2 - 2) and (3x + y^2 - 30) be a, b, c and d.

SUM of the four number =  x+ 17+ (3x - y^2 - 2) + (3x + y^2 - 30)=7x-15

As the four terms are consecutive terms of an AP => d-c = b-a ......... 3x+y^2-30-(3x-y^2-2)=17-x..........2y^2=45-x ......(i)
If y is an integer, x will be ODD, and so sum 7x-15 will be EVEN and divisible by 2.
If short of time, mark A. But let us find the value of x and y to confirm they are integers.

Also,  3x-y^2-2-(17)=17-x..........4x-36=y^2 .......(ii)

Substitute value of y^2 from ii in i.

2(4x-36)=45-x.......8x-72=45-x..........9x=117.......x=13

SUM = 7x-15 = 7*13-15 = 91-15 = 76 = 2*2*19
Hence, divisible by 1, 2, 4, 19, 38, and 76.

A

jigarfaria · Answer

The 4 consecutive terms can be written as:
(17-a), 17, (17+a), (17+2a) where 'a' is a common difference.

The sum is then given by,
Sum = (17-a) + 17 + (17+a) + (17+2a) = 17*4 + 2a = 2 * (17*2 + a).
Hence the sum is divisible by 2.

chetan2u · Answer

That is not correct method.

a or d need not be an integer.
say d is 1/2, then  2(2a+3d)=2(2a+\frac{3}{2})=4a+3 ....NOT divisible by 2 unless a is of the form odd/4.
This method could turn out to be a trap.
 16\frac{1}{2},17,17\frac{1}{2},18

The values of four terms should have told you to use these to get to your answer.

ravi1522 · Answer

Hello,
x, 17, (3x - y^2 - 2) and (3x + y^2 - 30) are four consecutive terms of an increasing arithmetic sequence 
so if 2nd term is 17 which is odd then first term and third term will be even so 
E+17+E+O so sum will be even 
Hence will be divisible by even no 
So option A is correct .

Thanks

Aish.0505 · Answer

Suppose one of the answer choices was a '4', then, this short cut method will not work?

x, 17, (3x - y^2 - 2) and (3x + y^2 - 30) are four consecutive terms

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

x, 17, (3x - y^2 - 2) and (3x + y^2 - 30) are four consecutive terms

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