nislam wrote:
In the sequence of positive integers \(a_1\), \(a_2\), \(a_3\),...,\(a_n−1\), \(a_n\) where \(n > 5\), each term after the first is \(d\) greater than the previous term, where \(d > 1\). What is the value of \(n\)?
1) \(a_1\) = 19 and \(a_n\) = 308
2) \(a_2\) = 36 and \(a_n−1\) = 291
There are a couple of ways of framing an approach this problem, but in the end, all of those ways will pass through divisibility. One way to frame it is algebraic. Expressed algebraically, the sequence goes as follows:
\(a_1\) = \(a_1\)
\(a_2\) = \(a_1\) +
d\(a_3\) = \(a_1\) + 2
d\(a_4\) = \(a_1\) + 3
d.
.
.
\(a_n\) = \(a_1\) + (
n - 1)
dWith an algebraic formula relating \(a_1\) to the other terms in the sequence, we can go to the statements and start plugging things in.
1) This statement gives us a value for \(a_1\) and for \(a_n\), meaning that the last line of our formula above can now be rewritten with a couple of numbers in it:
308 = 19 + (
n - 1)
d289 = (
n - 1)
dOne thing to note about this is that 289 is an unwieldy number, and we're expected to multiply two integers (
n - 1 and
d) to get that number. If you know your perfect squares up to at least \(17^2\), you're in luck! It turns out that 289 only has three factors: 1, 17, and 289 itself. We're told by the question stem that
d is greater than 1, and we're also told that
n is greater than 5, which means that the only way that
n - 1 and
d can multiply to 289 is if both are equal to 17. And if
n - 1 is equal to 17,
n itself must be 18.
SufficientPractically speaking, the only reason to bother to go to such trouble on this statement is that the difference between the terms is such a strange number. The GMAT has a history of making very hard DS questions that rely on the fact that numbers in those statements have limited divisibility. (For an example of this, see the classic
Official Guide problem involving $0.29 and $0.15 stamps.)
2) We can approach this statement by adapting our formula to the information provided and proceeding algebraically, as we did in (1). However, let's consider a different, more numerical approach, especially now that we've seen (1) boil down to an issue of factoring/divisibility. We can see that the difference between \(a_2\) and \(a_n-1\) here is 255. We also know that to get from \(a_2\) to \(a_n-1\), we have to add the same number
d repeatedly. Therefore, the question to ask is this: is there only one possibility for
d here (so that there'd also be only one possibility for the number of terms between the two given terms)? In this case, no: 255 breaks down into 3 x 5 x 17. This in turn means that
d could be 17 and that we add it 15 times to \(a_2\) to get to \(a_n-1\). It could also be that
d is 15 and that we must instead add it 17 times to get from \(a_2\) to \(a_n-1\). There are other possibilities (such as 51 increases of 5 each), but because we've already shown that \(n\) could have two different values, we can mark this statement
insufficient.
The answer is
A.