cat 2023 Question 21

Question

Let $a_n=46+8 n$ and $b_n=98+4 n$ be two sequences for natural numbers n ≤ 100. Then, the sum of all terms common to both the sequences is

Accepted Answer

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 lcm (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) ≤ 498

=> n is less than or equal to 50.5 => n = 50

Using the summation of A.P. formula:

Required sum = $\frac{n}{2}(2a + (n - 1)d) = \frac{50}{2}(2 imes 102 + 49 imes 8) = 14900$

Question 21.