CAT 2023 Slot 3 QA Question & Solution
AlgebraMedium
Question
Let $a_n = 46 + 8n$ and $b_n = 98 + 4n$ be two sequences for natural numbers $n \leq 100$. Then, the sum of all terms common to both the sequences is
Options
14900
14798
15000
14602
Solution
The first series goes as follows:
46, 54, 62, 70, 78, 86, 94, 102,....
The second series goes as follows:
98, 102, 106, 110,...
The first common term is 102 (first term of the common terms) and the common difference between them will be hcf(4,8) = 8
=> The required sequence is 102, 110, 118,..... (last term should be less than 468 (100th term of second series))
=> 102 + (n-1)(8) $\le$ 498
=> n is less than or equal to 50.5 => n = 50
Using the summation of A.P. formula:
Required sum = $\dfrac{n}{2}\left(2a+\left(n-1\right)d\right)=\dfrac{50}{2}\left(2\times\ 102+49\times\ 8\right)=14900$