Keep the square-root growth in the large-sample tail

$S=X_1+X_2+\cdots+X_{100}\sim N(300,400)$ and $T=X_{101}+X_{102}+\cdots+X_{400}\sim N(900,1200)$. The blocks are independent because they use disjoint independent observations.

$P(S>310)=0.309$ to 3 decimal places.

$E(3S-T)=3E(S)-E(T)=900-900=0$. $\operatorname{Var}(3S-T)=9\operatorname{Var}(S)+\operatorname{Var}(T)=9(400)+1200=4800$. Hence $3S-T\sim N(0,4800)$ and $P(3S>50+T)=P(3S-T>50)=0.235$ to 3 decimal places.

$N=X_1+\cdots+X_n\sim N(3n,4n)$. Thus $P(N>3.5n)=P(Z>(3.5n-3n)/\sqrt{4n})=P(Z>0.5n/(2\sqrt n))=P(Z>0.25\sqrt n)$.

As $n\to\infty$, $0.25\sqrt n\to\infty$, so the standard-normal upper-tail probability tends to 0. The source writes this informally as $P(Z>\infty)\to0$; precisely, the threshold tends to infinity and $P(Z>0.25\sqrt n)\to0$. The requested approximate probability for very large n is 0.