JEE 2023 — Mathematics PYQ

To find the sum of the common terms, we first need to identify the first common term, the common difference of the new sequence, and the last common term.

Step 1: Find the first common term ($a$)

Let's list a few terms of each sequence to find the first number that appears in all three:

• AP1: $3,7,11,15,19,23,27,31,35,39,43,47,\dots$

• AP2: $2,5,8,11,14,17,20,23,26,29,32,35,38,41,44,47,\dots$

• AP3: $2,7,12,17,22,27,32,37,42,47,\dots$

By inspection, the first term common to all three sequences is $47$.

So, $a=47$.

Step 2: Find the common difference ($d$)

The common difference of the resulting sequence of common terms is the Least Common Multiple (LCM) of the common differences of the three individual APs.

• Common difference of AP1 ($d_1$) $=7-3=4$

• Common difference of AP2 ($d_2$) $=5-2=3$

• Common difference of AP3 ($d_3$) $=7-2=5$

Step 3: Determine the number of terms ($n$)

The common terms must be less than or equal to the smallest last term of the three sequences. The limits are $399,359,$ and $197$. Therefore, the common terms cannot exceed $197$.

Using the general term formula $a_n=a+(n-1)d$:

Since $n$ must be an integer, the number of common terms is $n=3$.

Step 4: Calculate the sum

The common terms are:

1. $T_1=47$

2. $T_2=47+60=107$

3. $T_3=107+60=167$

Alternatively, using the sum formula $S_n=\frac{n}{2}[2a+(n-1)d]$:

Correct Option: (A) 321