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Sequence S is defined as follows: S 1 = 2

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Sequence S is defined as follows: S 1 = 2 [#permalink] New post 25 Jul 2020, 09:56
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Sequence S is defined as follows:\( S_1 = 2, S_2=2^1, S_3=2^2,......, S_n=2^{n-1}\). What is the sum of the terms is sequence S when \(n=10\) ?

A. \(2^9\)

B. \(2^{10}\)

C. \(2^{16}\)

D. \(2^{35}\)

E. \(2^{37}\)
[Reveal] Spoiler: OA

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Re: Sequence S is defined as follows: S 1 = 2 [#permalink] New post 25 Jul 2020, 10:55
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The question has a bug. You mention that Sn = 2^(n - 1), but from your description the first term is 2, which is 2^1, not 2^0. And also there isn't quite any correct answer either, because

S_n = a(r^n - 1)/(r - 1). Then S_n = 2(2^n - 1) or (2^n - 1) depending on what a is, and I don't think any of the options fit it.

Edit: missed that the first term is 2, not 1, welp. Answer is \(2^10\) then.

Last edited by Leaderboard on 27 Jul 2020, 11:03, edited 1 time in total.
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Re: Sequence S is defined as follows: S 1 = 2 [#permalink] New post 26 Jul 2020, 09:32
Leaderboard wrote:
The question has a bug. You mention that Sn = 2^(n - 1), but from your description the first term is 2, which is 2^1, not 2^0. And also there isn't quite any correct answer either, because

S_n = a(r^n - 1)/(r - 1). Then S_n = 2(2^n - 1) or (2^n - 1) depending on what a is, and I don't think any of the options fit it.


When n=10

S = S_1 + S_2 + .... + S_9 + S_10

=> S = 2 + 2^1 + 2^2 + 2^3 + .... + 2^8 + 2^9
=> S - 2 = 2^1 + 2^2 + 2^3 + .... + 2^8 + 2^9

Lets consider RHS as X

X = 2^1 + 2^2 + 2^3 + .... + 2^8 + 2^9
2*X = 2^2 + 2^3 + .... + 2^8 + 2^9 + 2^10

subtract first equation from second

=> 2*X - X = -2^1 + 0 + 0 + 0 + .... + 0 + 0 + 2^10
=> X = 2^10 - 2

S = X + 2 = 2^10
The answer is B.

It would have been better if it was mentioned that S_n = 2^(n-1) for n>1.
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Re: Sequence S is defined as follows: S 1 = 2 [#permalink] New post 27 Jul 2020, 11:02
Sum = 2^9 +2^8 +... + 2^2 + 2^1 + 2
Sum = 2(2^8 + 2^7 + ... +2 + 1 + 1)
Sum = 2^2(2^7 + 2^6 + ... + 2^1 + 1 + 1) ...
Sum = 2^10
Re: Sequence S is defined as follows: S 1 = 2   [#permalink] 27 Jul 2020, 11:02
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Sequence S is defined as follows: S 1 = 2

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