CAT 2023 Slot 2 QA Question 20: Let both the series $a_1, a_2, a_3, \ldots$ and $b_1, b_2, b_3, \ldots$ be in arithmetic progression such that the common differences of both the series are prime numbers. If $a_5 = b_9$, $a_{19} = b_

Question

Let both the series $a_1, a_2, a_3, \ldots$ and $b_1, b_2, b_3, \ldots$ be in arithmetic progression such that the common differences of both the series are prime numbers. If $a_5 = b_9$, $a_{19} = b_

Accepted Answer

Let the common differences of the two APs be $p$ and $q$ respectively, where $p$ and $q$ are prime numbers.

$a_{n} = A + (n - 1)p$
$b_{n} = B + (n - 1)q$

Since $b_2 = 0$:
$B + q = 0$
$B = - q$

Using $a_5 = b_9$:
$A + 4p = B + 8q$
$A + 4p = 7q$
$A = 7q - 4p$

Using $a_{19} = b_{19}$:
$A + 18p = B