The sequence a1,a2,,an, is such that an=(an1)(an3) for

IMO- Answer should be C. Please post OA.

Answer is C:

from stem: a4 =16

from 1: a1 = 2, from stem a4 = sqrt(2*a3) ==> 16 = sqrt(2*a3), so a3 = 128,
we just know the sequence as 2,a2,126,15,a5..., we need a5 to find a6, so NSF

from 2: a2 = 4, as a1 is not known, this statement itslef is insufficient NSF

combined 1+2:
a5 = sqrt(a4*a2) = 8,
sequence is like 2,4,128,16,8,..we have all values to find a6 so C

To Find a6 , we need to know the values of a5 & a3

Stmnt 1 - gives value of a3 - Not Suff
Stmtn 2 - gives value of a5 - Not suff

Combined - Suff

Ans - C

VERITAS PREP OFFICIAL SOLUTION:

One key for this problem, which looks worse than it actually is, is to avoid actually solving for the values. We only need to answer the question of whether a certain value can be solved for, not the more difficult question of what value would actually result.

The sequence rule gives that \(a_6=\sqrt{(a_5)(a_3)}\). In order to obtain the value of a6, we will need to obtain the values of both a5 and a3. Standing alone, a4 cannot produce either.

Adding statement (1) to the mix, we have the values of a1 and a4. Using the sequence rule with n=4 gives \(a_4=\sqrt{(a_3)(a_1)}\). Since we have both a4 and a1, this equation will allow us to solve for a3. From there, however, we can go no further. The values of a1,a3, and a4 do not collectively produce any of the other terms, and, in particular, we cannot solve for a5 with this information. Statement (1) alone is insufficient.

Taking statement (2) alone, we have the values of a2 and a4. Using the sequence rule with n = 5 gives \(a_5=\sqrt{(a_4)(a_2)}\). We have both a4 and a2, so we can get a5. Again, though, we reach a dead end. The values of a2,a4, and a5 do not collectively produce any of the other terms, and, in particular, we cannot solve for a3 with this information. Statement (2) alone is insufficient.

Combining the two statements, we are able to obtain all values from a1 through a5 (and beyond). In particular, statement (1) allows us to solve for a3, and statement (2) allows us to solve for a5. Recalling that \(a_6=\sqrt{(a_5)(a_3)}\), we know that we will be able to obtain a6. The two statements taken together are sufficient.

Bunuel,

Is this approach right.

√(ab)=√a√b ,

So a4= √a3*√a1
a5= √a4*√a2
a6= √a5*√a3
Weare given a4=16
a4= √a3*√a1

STMT 1: a1=2 .

16=√a3*√2

so √a3= 16/√2 = 8√2

STM2: a2=4

a5= √a4*√a2

a5=√16*√4

a5= 8

a6= √a5*√a3
From above we can find the value of a6

Only by knowing that...

a1 = 2
a3 = 128
a4 = 16

...can't we solve this using some kind of pattern recognition? I mean, all values in a sequence are related. Or are there more than one equation (string of operations?) that could possibly lie behind this pattern?

To get A6, we need A3 and A5. To get A5, we need A2 and A4.

1) A1=2
sqrt( 2 * A3) = 16
A3= 256
But this doesn't help us get A2, which means we can't get the A5 needed for A6.
Insufficient

2) A2 = 4
This helps us get A5, but A3 is still unaccounted for so we still can't get A6.
Insufficient.

1) helps us get A2, 2) helps us get A5, we have both of the numbers needed to get A6.

Answer: C

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