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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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