CAT 2023 Question Paper | Quant Slot 3

CAT 2023 Quant was dominated by Arithmetic followed by Algebra. In Arithmetic, the questions were dominated by topics like Speed-time-distance, Mixture and Alligations. This year, there was a surprise. The questions from Geometry were relatively on the lower side as compared to the previous years. There were 8 TITA Qs this year. Overall this section was at a medium level of difficulty.

Question 21 : 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

Video Explanation

Explanatory Answer

We have \( a _ { n } = 46 + 8 n \)and \( b _ { m } = 98 + 4 m \) where n, m are positive integers ≤ 100.
Let's find the smallest possible values of m , n where \( a _ { n } = b _ { m } \)
\( 46 + 8 n = 98 + 4 m \)
\( 8 n - 4 m = 52 \)
\( 2 n - m = 13 \)
\( n = 7 , m = 1 \)are smallest values of m, n where \( a _ { n } = b _ { m } \).
\( a _ { 7 } = b _ { 1 } = 102 \)
Therefore, the common elements in the series will be of the form \( 102 + 8 k \).
\( a _ { 100 } = 846 ; b _ { 100 } = 498 \)
The largest natural number less than or equal to 498 and of the form 102 + 8k is 494.
So the common terms of both the series are
102 , 110, 118, ..., 494
102, 102 + 8, 102 + 8(2), ..., 102 + 8(49)
So, there are 50 terms in this A.P
Sum of all the terms = \( \frac { 102 + 494 } { 2 } \times 50 = 14,900 \).

The question is " 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 "

Hence, the answer is '14900'

Choice B is the correct answer.

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