Bunuel wrote:
Asad wrote:
Bunuel wrote:
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.
Hello
BunuelIn 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.
From (1) we have that r = 10, and so the formula for the nth term is \(a_n = 10n\). Thus the sequence is \(a_1 = 10\), \(a_2 = 20\), \(a_3 = 30\), \(a_4 = 40\), \(a_5 = 50\), \(a_6 = 60\), \(a_7 = 70\), \(a_8 = 80\),
\(a_9 = 900\), \(a_{10} = 100\), ... 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"?Quote:
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.
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__