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QOTD#8 The first term in a certain sequence is 1, the 2nd

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QOTD#8 The first term in a certain sequence is 1, the 2nd [#permalink] New post 12 Sep 2016, 11:07
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77% (01:57) correct 22% (01:27) wrong based on 35 sessions
The first term in a certain sequence is 1, the 2nd term in the sequence is 2, and, for all integers n ≥ 3, the nth term in the sequence is the average (arithmetic mean) of the first n – 1 terms in the sequence. What is the value of the 6th term in the sequence?

Give your answer as a fraction.

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Question: 21
Page: 208-209


[Reveal] Spoiler: OA
3/2

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Re: QOTD#8 The first term in a certain sequence is 1, the 2nd [#permalink] New post 18 Nov 2016, 06:29
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Explanation


Before I answer this question, consider the following example:
Let's say that set T consists of {1, 2, 4, 5}. The AVERAGE = 3
What happens if we add 3 to get: {1, 2, 3, 4, 5}?
The average stays at 3 BECAUSE we added a value that was already the average of the original 4 values.

The same applies to this question:
term1 = 1
term2 = 2
term3 = (1+2)/2 = 1.5

term4: term 4 equals the AVERAGE of terms 1, 2 and 3. Notice that the AVERAGE of terms 1 and 2 is 1.5.
So, to find term4, we take terms 1 and 2 (which we already know has an average of 1.5) and we add to those values another 1.5 (which is term3), then the average won't change.
So, term4 = 1.5

And so on....
So, term6 = 1.5 and term7 = 1.5 and term8 = 1.5, and so on...

Answer:
[Reveal] Spoiler:
3/2


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Re: QOTD#8 The first term in a certain sequence is 1, the 2nd [#permalink] New post 16 Dec 2016, 00:14
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Can you please clarify. The question says "for all integers where n is equal to or greater than 3" the rule applies. If you take the third term, you can't apply that rule unless the third term is 3, right? If you take the average of 1 and 2, and average them, then you are following the rule but you are doing so for an integer that is less than the rule states, so it's a paradox. By my reading of the question, the correct answer should be 1,2,3, then you apply the rule, so the average of 1+2+3 is 3, then the average of 1+2+3+3 is 4.5, then the average of 1+2+3+3+4.5 is 6.75, and so on.
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Re: QOTD#8 The first term in a certain sequence is 1, the 2nd [#permalink] New post 14 Aug 2019, 00:44
This question is not clear to me, the above rule is applicable only for first 3rd number but in case of 4th number the result is not be 1.5, would you please elaborate the answer for 4th, 5th & 6th terms.
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Re: QOTD#8 The first term in a certain sequence is 1, the 2nd [#permalink] New post 14 Aug 2019, 04:37
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Question: The first term in a certain sequence is 1, the 2nd term in the sequence is 2, and, for all integers n ≥ 3, the nth term in the sequence is the average (arithmetic mean) of the first n – 1 terms in the sequence. What is the value of the 6th term in the sequence?

revengeoftheluddites wrote:
Can you please clarify. The question says "for all integers where n is equal to or greater than 3" the rule applies. If you take the third term, you can't apply that rule unless the third term is 3, right?


That's not correct.
The question says "for all integers n ≥ 3, the nth term in the sequence is...."
So, the rule applies to the 3rd term, the 4th term, the 5th term, etc.
In fact the rule applies to all terms after term 1 and term 2.

In other words, the part that says "for all integers n ≥ 3, the nth term in the sequence is...." is not referring to the VALUE of the terms. It's referring to the term number (e.g., term 5, term 11, etc)

Does that help?

Cheers,
Brent
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Re: QOTD#8 The first term in a certain sequence is 1, the 2nd [#permalink] New post 12 Nov 2019, 00:10
Re: QOTD#8 The first term in a certain sequence is 1, the 2nd   [#permalink] 12 Nov 2019, 00:10
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