What is the thousandth term of S, a certain sequence of numbers?

Target question: What is the value of term_1000?

Statement 1: For every n, the nth term of S is n².
Perfect! Statement 1 provides a "recipe" for determining the value of ANY term in the sequence.
So, it must be the case that term_1000 = 1000² = 1,000,000
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: The first five terms of S are 1², 2², 3², 4², and 5².
Important: Even though it certainly LOOKS like there is a pattern here, we can't assume that it continues,
For example, the sequence COULD just be a list of someone's favorite numbers.
So, it COULD be the case that term_6 = 17², term_7 = 5.3, term_8 = -9.22, etc, in which case, term_1000 can have pretty much any value
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer: A

Cheers,
Brent

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order does matter. Like a set, it contains members. The number of elements is called the length of the sequence.


(1) For every n, the nth term of S is n^2.
Clearly then 1000th term will be 1000^2.
Suff

(2) The first five terms of S are 1^2, 2^2, 3^2, 4^2, and 5^2.
Yes, there is 'some' order, but can we say exactly what with just 5 terms..NO. We require to know whether it is GP, AP or some pattern.
Here, it is possible that
a) the nth term is \(n^2\), so 1000thterm = \(1000^2\).
b) The sequence may be repetition of first 5 terms, so \(1^2, 2^2, 3^2, 4^2, 5^2,1^2, 2^2, 3^2, 4^2, 5^2......\). Then 100thterm will be \(5^2\).

A
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BrentGMATPrepNow ,Bunuel Please help.
I thought the answer was E because Statement 1 does not specify if n is an integer. So if n is an integer then it holds that the thousandth term will always be zero but if it not, it could be any number. Please help me clarify this thought as I'm still confused on why the answer is A

nth term means n is a positive integer. You cannot have n as 1/2, because 1/2th number is not possible. Order is always 1st, 2nd, 3rd and so on.

thousandth term will be 1000, so n is 1000
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There are many instances when integers are implied.
For example, if we're told that a classroom contains n children, we can conclude that n must be an integer.
Likewise, if we're talking about terms in a sequence, the total number of terms must be an integer.

(1) For every n, the nth term of S is n^2.
This definitely gives us the answer when we chose n=1000 the answer

Therefore clealry sufficient

(2) The first five terms of S are 1^2, 2^2, 3^2, 4^2, and 5^2.

We cannot definitely determine the rest of the series and how it will behave insufficient

Therefore IMO A

Hi IanStewart AndrewN
If in the question, it would have been mentioned that its a series instead of sequence, then would that make D the correct option. Because there is always a relationship in a series. In a sequence its not always true. Am I right?

Your definition of "series" isn't right. A "series" in math is a sum of a sequence. I've seen many prep companies use the word incorrectly (it's often used as if it were a synonym of "sequence", and it's not), but the GMAT will never use the word at all, since you learn about series in calculus, and calculus is beyond the scope of the test.

There is no mathematical word that means "a sequence with a relationship among terms", but even if we knew, in this question, that there was some relationship among the terms, the answer would still be A. You can't guess a relationship from a finite number of terms. Maybe the sequence in statement 2 is the sequence where the first five terms are 1, 4, 9, 16 and 25, and after that, every term is equal to the term five terms before it, so the sequence just repeats in a loop, 1, 4, 9, 16, 25, 1, 4, 9, 16, 25, etc. There are infinitely many other possibilities. You can only determine the thousandth term in a situation like this if you have an unambiguous rule that lets you calculate that term, so the answer here is A.
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Hello, pk6969. I did not think of any difference between series and sequence when I approached the question. If anything, the response that IanStewart gave above should show that rigid thinking about words when you are not absolutely sure of their meaning can get you into trouble. (To be fair to some prep companies and tutors, they may not know the mathematical definition of certain words and use them in a more general sense. But that does not mean that anybody, tutor, teacher, or prep company, should hide behind ignorance as an excuse not to improve on their understanding and teaching.)

Perhaps because I dabbled in intelligence testing in my teens and early twenties, statement (2) here looked rather familiar, like something on a "predict the next term" question. Trouble is, we are asked about the thousandth term, and it is a long way to go from the first five terms to the one in question. I like the example IanStewart gave in his post; my mind went straight to a different type of sequence:

Given: 1^2, 2^2, 3^2, 4^2, 5^2

Possible: 1^3, 2^3, 3^3, 4^3, 5^3

Without a way to pin down the behavior of the sequence as a whole, I knew the proposition was dead in the water. (A) made sense; (B) (and, by extension, (D)) did not. I talk sometimes about the power of logic in my once-in-a-while Quant posts. These types of questions are my bread and butter, requiring little in the way of mathematical acumen to answer accurately and efficiently. (Again, I do not mean to excuse myself from improving my mathematical understanding. I am working on it all the time.)

- Andrew
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if we are saying " a sequence "and giving first five terms which has difference of 2...then how can at 6th term we say it can again a loop 1^2,2^2......

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