Compare arithmetic and geometric partial sums

With AP terms $a+6d,a+2d,a$ forming the GP, $(a+2d)^2=a(a+6d)$. Thus $2d(2d-a)=0$, and $d\ne0$ gives $a=2d$. The GP ratio is $a/(a+2d)=1/2$, so its series converges.

$a+6d=8d=3$, so $d=3/8$, $a=3/4$, while the GP starts at $3$. The difference is $D_n=\dfrac{3n(n+3)}{16}-6(1-2^{-n})$. Require $D_n\ge100$.

The source table gives $D_{22}\approx97.125$, $D_{23}\approx106.13$, $D_{24}\approx115.5$. Moreover $D_{n+1}-D_n=3/4+3n/8-3/2^n>0$ for $n\ge2$, and the first two differences are below $100$. Thus the smallest value is $n=23$.