CAT 2023 Slot 3 — QA Question 20

List the two sequences for $1\le n\le 100$ (note $a_n$ uses $n\le 99$ to stay within range too, but the overlap is what matters):

$$\begin{aligned} &a_n=46+8n:\ 54,\,62,\,70,\,\dots,\,838\quad(d=8)\\ &b_n=98+4n:\ 102,\,106,\,110,\,\dots,\,498\quad(d=4) \end{aligned}$$

Common difference of the overlap $=\operatorname{lcm}(8,4)=8$. The first term common to both is $102$. So the common terms are:

$$102,\,110,\,118,\,\dots\quad(d=8)$$

Find how many common terms there are. They cannot exceed $b_n$'s top value $494$ (the largest term of $b_n$ that is also $\le 838$); solving $102+8(k-1)\le 494$:

$$\begin{aligned} &102+8(k-1)\le 494\ \Rightarrow\ 8(k-1)\le 392\\ &\Rightarrow\ k-1\le 49\ \Rightarrow\ k\le 50 \end{aligned}$$

So there are $50$ common terms, last one $=494$.

Sum the AP of $50$ terms:

$$S=\frac{50}{2}\,(102+494)=25\times 596=14900$$