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655-705 Level|   Sequences|      
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Bunuel
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The sequence of numbers p, ps, ps^2, ps^3, ps^4, ps^5 is a geometric 05 Apr 2022, 02:15
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The sequence of numbers \(p, \ ps, \ ps^2, \ ps^3, \ ps^4, \ ps^5\) is a geometric progression. The sum of the first three terms in the series is one-ninth the sum of the first six terms. What of the ratio of the fifth term to the second term?

(A) 1:1
(B) 2:1
(C) 3:1
(D) 8:1
(E) 9:1
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D
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divyadna
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The sequence of numbers p, ps, ps^2, ps^3, ps^4, ps^5 is a geometric 09 Apr 2022, 09:17
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neeleshg06
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The sequence of numbers p, ps, ps^2, ps^3, ps^4, ps^5 is a geometric 19 Aug 2024, 04:49
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Can someone please share an explanation for this question.
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thecardinal
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The sequence of numbers p, ps, ps^2, ps^3, ps^4, ps^5 is a geometric 22 Aug 2024, 07:19
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neeleshg06
Can someone please share an explanation for this question.
­yeappp!!
so what I did is to get out of my mind extracting an analytical answer and started applying numbers.. Intuitively you can catch that you wont need to go pretty far with integers to get to 1/9
so I applied s=1 and (1 + s + s^2) / (1 + s + s^2 + s^3 + s^4 + s^5) = 1/2 not the case
then s = 2 and (1 + s + s^2) / (1 + s + s^2 + s^3 + s^4 + s^5) = 1/9. that's it
so s^4/s=16/2=8
so 8:1
I hope this helpss!!­
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The sequence of numbers p, ps, ps^2, ps^3, ps^4, ps^5 is a geometric 28 Aug 2024, 21:43
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­They wanted the ratio of ps^4 / ps which equals s^3/1
The only cubes present in the options are 8, so 8:1 and 1:1 so that narrows down our options.

If the answer was 1:1, it means s was 1, which would mean it is one number repeating over and over again. However, we see that the sum of terms are not such, they aren't one number repeating over and over again. That only leaves us with 8:1.
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The sequence of numbers p, ps, ps^2, ps^3, ps^4, ps^5 is a geometric 22 Oct 2024, 17:12
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The sequence of numbers \(p, \ ps, \ ps^2, \ ps^3, \ ps^4, \ ps^5\) is a geometric progression. The sum of the first three terms in the series is one-ninth the sum of the first six terms. What of the ratio of the fifth term to the second term?

The sum of the first three terms in the series is one-ninth the sum of the first six terms:

\((p + ps + ps^2) = \frac{1}{9}(p + ps + ps^2 + ps^3 + ps^4 + ps^5)\)

\((ps^3 + ps^4 + ps^5) = \frac{8}{9}( p + ps + ps^2 + ps^3 + ps^4 + ps^5)\)

\(\frac{(p + ps + ps^2)}{(ps^3 + ps^4 + ps^5)} = \frac{1}{8}\)

\(\frac{1}{s^3} = \frac{1}{8}\)

\(8 = s^3\)

Ratio of the fifth term to the second term:

\(\frac{ps^4}{ps} = \frac{s^3}{1} = 8:1\)

(A) 1:1
(B) 2:1
(C) 3:1
(D) 8:1
(E) 9:1

Correct answer: D
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The sequence of numbers p, ps, ps^2, ps^3, ps^4, ps^5 is a geometric 22 Oct 2024, 20:04
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hey marty. genuine question, how do i start to think like this? what made you think that using the sum of the rest of the terms would give this answer?
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MartyMurray
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The sequence of numbers p, ps, ps^2, ps^3, ps^4, ps^5 is a geometric 23 Oct 2024, 06:10
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samriddhi1234
hey marty. genuine question, how do i start to think like this? what made you think that using the sum of the rest of the terms would give this answer?
In general, I start looking for a connection between a question and something I'm familiar with or a pattern or some kind of logic to what's presented.

In this case, we know that the terms form a geometric sequence. So, we have the pattern of their increasing by being multiplied by s repeatedly.

Then, I often just try something.

In this case, since the first three terms are part of the first six terms, I tried seeing what would happen if I separated them into 1/9 and 8/9.

At that point, I was set since the first three terms and the remaining terms are equal in number and I saw that the last three were a multiple of the first.
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