JEE Main 2020 (Online) 9th January Evening ShiftQuestion 1
The number of terms common to the two A.P.'s $3,7,11,\ldots,407$ and $2,9,16,\ldots,709$ is ________.
Sequence and Series · Arithmetic Progression
Answer$14$
Solution
Formulas used. (1) Common terms of two A.P.s form an A.P. with difference $\operatorname{lcm}(d_1,d_2)$(2) $T_n=a+(n-1)d$Step 1 - First common term. First A.P.: remainder $3$ on division by $4$. Second A.P.: $2,9,16,23,\dots$ (remainder $2$ by $7$). Checking $2,9,16,23$: $23$ leaves remainder $3$ on division by $4$. So the first common term is $23$. Step 2 - Common A.P. Difference $\operatorname{lcm}(4,7)=28$: $23,51,79,\dots$ Step 3 - Count. Terms must not exceed $\min(407,709)=407$: $$23+28k\le407\ \Longrightarrow\ k\le13.7,\quad k=0,1,\dots,13 .$$ There are $14$ common terms. Answer: 14