JEE Main 2024 (Online) 27th January Morning ShiftQuestion 1
The number of common terms in the progressions $4,9,14,19,\ldots$, up to $25^{\text{th}}$ term and $3,6,9,12,\ldots$, up to $37^{\text{th}}$ term is :
Sequence and Series · Arithmetic Progression
Solution
Formulas used. (1) $a_n=a+(n-1)d$(2) Common terms of two A.P.s form an A.P. with difference $\operatorname{lcm}(d_1,d_2)$Step 1 - Last terms. First A.P.: $4+24(5)=124$. Second A.P.: $3\times37=111$. A common term is at most $111$. Step 2 - First common term. The first A.P. has terms that leave remainder $4$ on division by $5$; the second has multiples of $3$. The first number of both kinds is $9$. Step 3 - The common A.P. Difference $\operatorname{lcm}(5,3)=15$: $$9,\ 24,\ 39,\ 54,\ 69,\ 84,\ 99\quad(\text{next is }114>111).$$ There are $7$ common terms. Answer: (B)