If each term in an infinite sequence is found by the equation Sx=8x–6,

Question

If each term in an infinite sequence is found by the equation  S_x=8^x-6 , and  S_1=2 , is every term in the sequence divisible by y?

(1) y is an even integer.

(2) At least 2 of the first 4 terms in the sequence are divisible by y.

Kudos for a correct  solution .

Bunuel · Accepted Answer

VERITAS PREP OFFICIAL SOLUTION:

The correct response is (B). If you could determine the value of y, you could say whether every term in the sequence S is divisible by y. Start by expanding the sequence: 2, 58, 506, 4090,…etc. The only way that y can be a factor of ALL the terms is if y = 1 or y = 2.

Statement (1) is not sufficient. If y = 2, then the answer is yes, but if y equals a different even number such as 4, then the answer is no.

Statement (2) tells us that at minimum, two terms are divisible by y. If y = 1 or y = 2, then this holds true and the answer is yes. There is no other common factor of any two of 2, 58, 506, 4090…, etc, so y must equal either 1 or 2. Sufficient.

chetan2u · Answer

hi Bunuel, is something missing or typed wrongly in eq Sx..

Bunuel · Answer

Edited formatting error. Thank you.

gmat6nplus1 · Answer

S(1)=2;  S(2)=2^6-6 ;  S(3)=2^9-6 ;  S(4)=2^1^2-6

Statement 1: if y = 2, -2, 0 then every term is a multiple of y. If y = 4 then S(1) becomes a proper fraction. Not sufficient.

Statement 2: pick two random terms among those listed above. Each term is divisible by 2, -2, 1, -1, and a multiple of 0 and these values will be divisible by every other number in the senquence.

Answer B.

*Edited: added "a multiple of" before 0.

DesiGmat · Answer

Here we go:

S_x=8^x-6, and S_1=2 (Given)
 
St1: y is an even integer

Well the sequence will always be divisible by 2.
but if we take y as 4 or 6, sequence will not be divisible
Clearly not sufficient.

St2: At least 2 of the first 4 terms in the sequence are divisible by y.

we need to consider the first 4 terms, so better to find out the terms in order to be accurate..

First term = 2 (given) (X = 1)
Second term = 58 = 2*29
Third term = 506 = 2*11*23
Fourth term = 4090 = 2*5*409

Randomly consider any 2 terms.
Common factor : 1, 2, -1, and -2

Option B is correct

isoyeon · Answer

If each term in an infinite sequence is found by the equation S(x)=8^x – 6, and S(1)=2, is every term in the sequence divisible by y?

(1) y is an even integer.

(2) At least 2 of the first 4 terms in the sequence are divisible by y.

(This is a question from Veristasprep. It gave me an explanation for the answer, but I still can't seem to understand the reasoning. Hoping someone can explain the answer. Thank you!!!!)

sillyboy · Answer

B.

S(1) = 2
S(2) = 58 = 2*29
S(3) = 8^3 - 6 = 2^9 - 6 = 506 = 2*11*23
S(4) = 8^4 - 6 = 2^12 - 6 = 4090 = 2*5*409

Statement 1: y is an even integer
Y can be any one of 2,4,6,.......NOT SUFFICIENT

Statement 2: At least 2 of the first 4 terms in the sequence are divisible by y.
The only common factor is 2. So, Y = 2. SUFFICIENT

Bunuel · Answer

Merging topics. Please refer to the discussion above.

Aashav27 · Answer

Why are we assuming x to be an integer? It is not mentioned in the question. If we assume S(1.1), then none of the statements will be sufficient, correct?

chetan2u · Answer

x is the place of that term  S_x  In the sequence. 
So it cannot be a fraction. The place of a term has to be 1st, 2nd and so on. 
x=1,2,3..

snscatter · Answer

Hello,

Please help me rectify my approach:

S 1  =  8^1-6  = 2
S 2  =  8^2 - 6  = 10
S 3  =  8^3 - 6  = 506
S 4  =  8^4 - 6  = 4090

is every term in the sequence divisible by y?

1) y is an even integer. 
y can be 2,4,6,-2,-4 etc

Here S1 = 2 
Only 2 or -2 can be divisors of S1 because y is even
Thus y can be 2 or -2

2) At least 2 of the first 4 terms in the sequence are divisible by y.
Taking permutations of two terms and their CFs: 
(2,10) --> (-1,-2,1,2), 
(2,506) --> (-1,-2,1,2), 
(10,4090) --> (-1,-2,-10,1,2,10),
(10,506) --> (-1,-2,1,2) etc
 Thus, y can be any of the following -> -1,-2,-10,1,2,10
But only (-1,-2,1,2) can be divisors of more than 2 terms out of these 4 terms

Now both options seem sufficient but Statement(2) gives me a bigger list of y. So answer is (B)

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If each term in an infinite sequence is found by the equation Sx=8x–6,

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