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Question of the Week- 18 (Sum of first 11 terms of an AP . . . . . )

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Question of the Week- 18 (Sum of first 11 terms of an AP . . . . . )  [#permalink]

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New post Updated on: 12 Oct 2018, 05:04
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Question Stats:

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e-GMAT Question of the Week #18

Sum of first 11 terms of an arithmetic progression is 275. Which of the following can be the sum of the first 12 terms, if the first term and the common difference are positive integers?

    A. 300
    B. 310
    C. 316
    D. 324
    E. 333

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Originally posted by EgmatQuantExpert on 12 Oct 2018, 04:10.
Last edited by EgmatQuantExpert on 12 Oct 2018, 05:04, edited 1 time in total.
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Re: Question of the Week- 18 (Sum of first 11 terms of an AP . . . . . )  [#permalink]

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New post 12 Oct 2018, 04:44
For sum of first 11 terms of the progression
we can use the formula:
n/2(2a+(n-1)d)
which will give us the value
50=2a+10d

also sum of progression is:
n/2[first term + nth term)

by that:
a1+a11=50
since it is mentioned in the question that the difference common difference d and the first term should be an integer then we can conclude that a1=5 and a11= 45 and the common difference d= 4 so the next term ie. a12 should be 49 so our answer here is 275+49=324
answer choice : D


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Re: Question of the Week- 18 (Sum of first 11 terms of an AP . . . . . )  [#permalink]

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New post 13 Oct 2018, 23:32
Since we are given, the sum of first 11 terms of an arithmetic progression is 275. We can take the first term to be 'a' and the subsequent terms with a common difference of 'd', added to the previous term in the sequence.

Hence, the terms would be {a, a+d, a+2d, ..., a+10d} when the series has 11 terms in it.

Similarly, when the series would be having 12 terms, the relevant terms would be {a, a+d, ..., a+11d}

Utilizing the formula of AP, we know sum of 'n' terms would be S = n/2 * [2a + (n-1) * d]

Substituting n = 11, we get the following.

S = 11/2 [2a + 10d] = 275

Simplifying this equation, we get the value of a+5d to be 25.

Now, since we are being asked to determine the sum of 12 terms in the series, we can represent it as the following.

(Sum of 11 terms in the series + 12th term)

So, this would be (275 + a + 11d). Since we know, the value of a + 5d is 25. We can represent it as, 300 + 6d.

Hence, we know the sum of 12 terms in the series would be more than 300 and also the common difference would be a multiple of 6 (it is also given that the common difference is a positive integer).

Reviewing the answer choices given, the answer would be 'D'.
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Re: Question of the Week- 18 (Sum of first 11 terms of an AP . . . . . )  [#permalink]

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New post 17 Oct 2018, 00:52

Solution


Given:
    • The sum of first 11 terms of an arithmetic progression = 275
    • First term and common difference are positive integers

To find:
    • The sum of the first 12 terms

Approach and Working:
    • Let us assume that the first term of the arithmetic progression is ‘a’ and common difference is ‘d’
    • Sum of first 11 terms, \(S_{11}\) = a + (a + d) + (a + 2d) + (a + 3d) + …. + (a + 9d) + (a + 10d)
      o Implies, \(S_{11}\) = 11a + d * (1 + 2 + 3 + 4 + … + 9 + 10) = 11a + d * 10 * 11/2 = 11a + 55d
      o We are given, \(S_{11}\) = 275 = 11a + 55d
      o Thus, a + 5d = 25 ……… (1)

    • We can write the sum of first 12 terms as \(S_{12}\) = \(S_{11}\) + \(12^{th}\) term
      o Implies, \(S_{12}\) = 275 + (a + 11d) = 275 + (a + 5d) + 6d = 300 + 6d
      o We are given that d is a positive integer

    • Now, if we observe the answer choices, the only option which is in the form of 300 + 6d is 324

Therefore, \(S_{12}\) can be equal to 324

Hence the correct answer is Option D.

Answer: D

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Re: Question of the Week- 18 (Sum of first 11 terms of an AP . . . . . )   [#permalink] 17 Oct 2018, 00:52
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