CAT 2023 Slot 3 QA Q21 Progressions & Series

Question 21 of the Quantitative Ability (QA) section, CAT 2023 Slot 3, with CAT's official answer and a worked solution.

Let \(a_n=46+8 n\) and \(b_n=98+4 n\) be two sequences for natural numbers \(n \leq 100\). Then, the sum of all terms common to both the sequences is
  1. A
    15000
  2. B
    14900
  3. C
    14602
  4. D
    14798

Answer: (B) 14900 Progressions & Series MCQ

Solution

  1. \(a_n\) (n = 1 to 100): 54, 62, ..., 846, all \(\equiv 6 \pmod 8\). \(b_n\): 102, 106, ..., 498.
  2. \(b_n = 98+4n \equiv 6 \pmod 8\) exactly when n is odd: 102, 110, ..., 494 (n = 99). Note 498 (n = 100) is \(\equiv 2\pmod 8\), so it is not common.
  3. All these lie in the range of \(a_n\). Count \(= \frac{494-102}{8}+1 = 50\); sum \(= 50\times\frac{102+494}{2} = 14900\).

Same concept in other years

See every question on Common terms of two APs form an AP with difference = LCM.

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