Bunuel wrote:
If \(a_n=f(n)\) for all integers n>0, is \(a_4>a_8\)?
(1) \(a_5=13\)
(2) f(x)=9−f(x−1) for all integers x
Kudos for a correct solution.
VERITAS PREP OFFICIAL SOLUTION:There’s very little information in the question stem, so we turn directly to statement (1). Without an equation for f(n), we have no rule to relate the different terms of the sequence. The value of a5 is therefore insufficient standing alone.
Turning to statement (2), the temptation is to assume insufficiency because no term of the sequence is given or can be determined. And of course the two statements will be sufficient taken together, since they will collectively determine every term of the sequence.
Not so fast, though! Things should perhaps feel a bit “too easy” at this point, so let’s explore statement (2) alone a bit further. Combining statement (2) with the question stem gives \(a_n=f(n)=9-f(n-1)=9-a_{n-1}\).
Now, plug in a value for n – say, for instance, n=5. The equation becomes a5=9−a4. Similarly, we obtain a6=9−a5. Finally, substitute for \(a_5:a_6=9-(9-a_4)=9-9+a_4=a_4\). By the same token, a8=a6. It turns out that, regardless of actual values, every other term of the sequence is equal. And since a4=a6=a8, the answer to the question “is a4>a8?” is “no.” So statement (2) is sufficient standing alone.
(1) Says a5 is 13. If we substitute this in the equation,
But according to your explanation each term is equal. Arent the two statements contradicting each other?