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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]  12 Sep 2016, 11:07
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Question Stats:

71% (01:49) correct 28% (01:39) wrong based on 7 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?

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[Reveal] Spoiler: OA
3/2

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Sandy
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Kudos [?]: 1613 [0], given: 375

Re: QOTD#8 The first term in a certain sequence is 1, the 2nd [#permalink]  12 Sep 2016, 11:16
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Explanation

For all integers n ≥ 3, the nth term in the sequence is the average of the first n – 1 terms in the sequence.

The 3rd term, which is the average of the first 2 terms, is $$\frac{(1+2)}{2}$$, or 1.5.

The 4th term, which is the average of the first 3 terms, is $$\frac{(1+2+1.5)}{3}$$, or 1.5.

Similarly, the 5th term is $$\frac{(1+2+1.5+1.5)}{4}= \frac{6}{4}=1.5$$, and the 6th term is $$\frac{(1+2+1.5+1.5+1.5)}{5}$$ or 1.5. Since you must give your answer as a fraction, the correct
answer is $$\frac{3}{2}$$.
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Re: QOTD#8 The first term in a certain sequence is 1, the 2nd [#permalink]  18 Nov 2016, 06:29
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Expert's post
sandy wrote:
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?

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...

[Reveal] Spoiler:
3/2

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Re: QOTD#8 The first term in a certain sequence is 1, the 2nd [#permalink]  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.
Re: QOTD#8 The first term in a certain sequence is 1, the 2nd   [#permalink] 16 Dec 2016, 00:14
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