If r is a constant a_n = rn for all positive integers n, for how many

Since \(a_n = rn\), then \(a_{100} =100r\) and \(a_{105} =105r\)

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

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 \(a_n < 100\)?

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?

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.

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.

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__

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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