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# A certain sequence is defined by the following rule: Sn = k(Sn–1), whe

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Math Expert
Joined: 02 Sep 2009
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A certain sequence is defined by the following rule: Sn = k(Sn–1), whe  [#permalink]

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12 Jun 2015, 03:10
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A certain sequence is defined by the following rule: $$S_n = k(S_{n-1})$$, where k is a constant. If S1 = 64 and S25 = 192, what is the value of S9 ?

(A) $$\sqrt{2}$$

(B) $$\sqrt{3}$$

(C) $$64\sqrt{3}$$

(D) $$64\sqrt[3]{3}$$

(E) $$64\sqrt[24]{3}$$

Kudos for a correct solution.

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Re: A certain sequence is defined by the following rule: Sn = k(Sn–1), whe  [#permalink]

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12 Jun 2015, 03:46
1
D.
S25 = k^24*S1
ie k = 24th root of 3
S9 = k^8*S1
ie S9 = cube root 3 x 64 = Option D
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Re: A certain sequence is defined by the following rule: Sn = k(Sn–1), whe  [#permalink]

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12 Jun 2015, 04:11
1
3
Bunuel wrote:
A certain sequence is defined by the following rule: $$S_n = k(S_{n-1})$$, where k is a constant. If S1 = 64 and S25 = 192, what is the value of S9 ?

(A) $$\sqrt{2}$$

(B) $$\sqrt{3}$$

(C) $$64\sqrt{3}$$

(D) $$64\sqrt[3]{3}$$

(E) $$64\sqrt[24]{3}$$

Kudos for a correct solution.

$$S_n = k(S_{n-1})$$

i.e. $$S_2 = k(S_{2-1}) = k(S_1)$$
and $$S_3 = k(S_{3-1}) = k(S_2) = k^2*(S_1)$$
and $$S_4 = k(S_{4-1}) = k(S_3) = k^3*(S_1)$$
and $$S_5 = k(S_{5-1}) = k(S_4) = k^4*(S_1)$$
...
...
and $$S_{25} = k(S_{25-1}) = k(S_24) = k^{24}*(S_1)$$

i.e. $$S_{25} = 192 = k^{24}*64$$
i.e. $$3 = k^{24}$$
i.e. $$3^{1/3} = k^8$$

Now, $$S_9 = k^8*(S_1) = 3^{1/3}*64$$

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Re: A certain sequence is defined by the following rule: Sn = k(Sn–1), whe  [#permalink]

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15 Jun 2015, 02:35
Bunuel wrote:
A certain sequence is defined by the following rule: $$S_n = k(S_{n-1})$$, where k is a constant. If S1 = 64 and S25 = 192, what is the value of S9 ?

(A) $$\sqrt{2}$$

(B) $$\sqrt{3}$$

(C) $$64\sqrt{3}$$

(D) $$64\sqrt[3]{3}$$

(E) $$64\sqrt[24]{3}$$

Kudos for a correct solution.

MANHATTAN GMAT OFFICIAL SOLUTION:

To form each new term of the sequence, we multiply the previous term by k. Given that S1 = 64, we know that S2 = 64k, and S3 = 64k^2, and Sn = 64k^(n–1). Thus S25 = 64k^24 . Since we are toldthat S25 = 192, we can set up an equation to solve for k as follows:

64k^24 = 192
k^24 = 3
k = 3^(1/24)

Plugging this value for k into the expression for S9, we have:

$$S_9=64k^8=64*(3^{(\frac{1}{24})})^8=64*3^{\frac{1}{3}}=64\sqrt[3]{3}$$

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A certain sequence is defined by the following rule: Sn = k(Sn–1), whe  [#permalink]

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14 Dec 2016, 07:20
Sn = k * (Sn-1)
S1 = 64
S25 = 192

Now $$k = (192/64)^(\frac{1}{24}) = 3^(\frac{1}{24})$$

Now $$S9 = S1 * k^8 = 64 * 3^(\frac{1}{24})^8 = 64 * 3^(1/3) = 64 * 3\sqrt{3}$$

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Re: A certain sequence is defined by the following rule: Sn = k(Sn–1), whe  [#permalink]

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30 Jul 2018, 01:04
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Re: A certain sequence is defined by the following rule: Sn = k(Sn–1), whe &nbs [#permalink] 30 Jul 2018, 01:04
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