If r is a constant and \(a_n = rn\) for all positive integers n, for how many values of n is \(a_n < 100\)?

(1) \(a_{50} = 500\):

Since \(a_n = rn\), then \(a_{50} =r*50= 500\). Hence, r = 10. We know everything about the sequence, so we can answer the question. Sufficient.

(2) \(a_{100} + a_{105} = 2050\):

\(100r + 105r = 2050\) --> r = 10. The same here. Sufficient.

Answer: D.
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Hi Bunuel,

Thanks for the explanation.

But how did you come up with

a100+a105=2050 => 100r+105r=2050?

Since \(a_n = rn\), then \(a_{100} =100r\) and \(a_{105} =105r\)
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Thanks so much. It's clear now.

Just simply replace a100 with 100r and a105 with 105r => 100r+105r=2050 => 205r=2050 => r=10.

Bunuel

Hi, I just have one confusion here.

The question has not specified anything regarding the construction of the equation. How did you assume that a50 = r*50?

As we can also assume r = 5 and a50 = r*25. Can you please help me understand the same.

Addressed here: https://gmatclub.com/forum/if-r-is-a-co ... l#p2371870
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Bunuel

My question was on this arrival - Since an=rnan=rn, then =500a50=r∗50=500

Isn't it given in the stem?

If r is a constant and\(a_n = rn\) for all positive integers n, for how many values of n is \(a_n < 100\)?
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Bunuel

Sorry! I am still not clear. Cant r (a constant value) be anything such as 5 or 15 or (may be 10)

The stem says: \(a_n = r*n\)

If r is a constant and \(a_n = rn\) for all positive integers n, for how many values of n is \(a_n < 100\)?

(1) says: \(a_{50} = 500\). Since \(a_n = rn\), then \(a_{50} =r*50= 500\). \(r*50= 500\). Hence, r = 10.

Can you please tell me which step there is unclear?
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Hello Bunuel
In this question, r is constant. So, should we find out the value of r?
In statement 1 and statement 2, we have the value of n (whatever the value of n is, we can definitely know either \(a_n < 100\) or not).
Is not it?
One more thing:
What if the statement 2 is like below:
(2) \(a_{100} × a_{20} = 200000\)
Is it still sufficient?
My calculation says: the 'r' is +/- 10.

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I think you misunderstood the question.

n there is an index number. \(a_n\) is the nth term in the sequence. \(a_{50} = 500\) does not mean that we have an answer: \(a_{50} = 500>100\). That's not what the question asks. The question asks, for how many values of n is \(a_n < 100\). This translates to: how many terms of the sequence are less than 100.

From (1) we have that r = 10, and so the formula for the nth term is \(a_n = 10n\). Thus the sequence is \(a_1 = 10\), \(a_2 = 20\), \(a_3 = 30\), \(a_4 = 40\), \(a_5 = 50\), \(a_6 = 60\), \(a_7 = 70\), \(a_8 = 80\), \(a_9 = 90\), \(a_{10} = 100\), ... Therefore, the answer to the question is: for 9 values of n is \(a_n < 100\) or 9 terms of the sequence are less than 100.

If (2) were \(a_{100} × a_{20} = 200000\), then 100r*20r = 200000 --> r = 10 or r = -10. If r = 10, then the answer is 9 (see above) and if r = -10, then the sequence is -10, -20, -30, ... so all terms of the sequence will be less than 100. So, (2) would not be sufficient in this case.

Hope it helps.
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Thanks Bunuel for giving time in my post(s).
My message:
The question says that "r" is constant. So, it is specific (it could be 10, -10, 10000, 10000000, anything but just one specific thing-not more than one specific value). I mean it could not be one figure at a time.
I've made the newly creative example so that it confirms 2 constant values e.g., 10, -10 (which is not possible in real life, possibly) of 'r'. But, in statement 1, r is just 10 (not -10 anymore). So, it seems that the statement 1 (r=10) and my creative one (r=10, -10) contradict each other that is not possible in official question. So, it confirms that my creative statement is not valid, i guess. In your explanation, you've tried to find out the value of r (10) in both statements. My question: IF it is constant WHY do we try to find out the value of "r"?


I know that. I know it is 9th term for sure. If it is ≥ 10th term, it does not satisfy our question prompt. But, my message was different-the message is explained before this "quoting" part.


PS-There is a typo in newly highlighted part. It is better if you edit it. Thanks__
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Sorry but I don't understand what you are trying to say. We need r to answer the question. Different values of r give different answers to the question. Here the answer is D because each statement is sufficient to get r and answer the question. IF (2) were \(a_{100} × a_{20} = 200000\), then the question would still be valid. The answer would be A in that case and not D, because from (2) we get two values of r and two different answers to the question (9 and ALL). Also, the statement would not contradict each other we'd have r = 10 from (1) (sufficient) and r = 10 or r = -10 from (2) (not sufficient).
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