The number of common terms in the progressions 4, 9, 14, 19, , up...

$4, 9, 14, 19, \ldots$, up to ${25}^{\text{th}}$ term

$⇒ {T}_{25} = 4 + ( 25 - 1 ) 5$

$⇒ {T}_{25} = 4 + 120$

$⇒ {T}_{25} = 124$

$3, 6, 9, 12, \ldots$, up to ${37}^{\text{th}}$ term

$⇒ {T}_{37} = 3 + ( 37 - 1 ) 3$

$⇒ {T}_{37} = 3 + 108$

$⇒ {T}_{37} = 111$

Common difference of ${I}^{\text{st}}$ series is ${d}_{1} = 5$

Common difference of ${II}^{\text{nd}}$ series is ${d}_{2} = 3$

First common term $= 9$, and their common difference $= 15 ( LCM of {d}_{1} and {d}_{2}).</p><p>So, the common terms are$9, 24, 39, 54, 69, 84, 99$.</p>$