$b_1, b_2, b_3, \ldots \ldots \ldots \ldots, b_{101} \text { are in G.P. }$
$\text { Given }: \log _e\left(b_2\right)-\log _e\left(b_1\right)=\log _e(2) \Rightarrow \frac{b_2}{b_1}=2=r \text { (common ratio of G.P.) }$
$a_1, a_2, a_3, \ldots \ldots \ldots a_{101} \text { are in A.P. }$
$a_1=b_1=a$
$b_1+b_2+b_3+\ldots \ldots \ldots b_{51}=t,$
$S=a_1+a_2+\ldots \ldots+a_{51}$
$t=\text { sum of } 51 \text { terms of G.P. }=b_1 \frac{\left(r^{51}-1\right)}{r-1}=\frac{a\left(2^{51}-1\right)}{2-1}=a\left(2^{51}-1\right)$
$\left.\left.s=\text { sum of } 51 \text { terms of A.P }=\frac{51}{2}\right] 2 a_1+(n-1) d\right]=\frac{51}{2}(2 a+50 d)$
$\text { Given } a_{51}=b_{51}$
$a+50 d=a(2)^{50}$
$50 d=a\left(2^{50}-1\right)$
$\text { Hence, } \Rightarrow s=a\left(51.2^{49}+\frac{51}{2}\right)$
$s=2\left(4.2^{49}+47.2^{49}+\frac{51}{2}\right) \Rightarrow$ $s=a\left(\left(2^{51}-1\right)+47.2^{49}+\frac{53}{2}\right)$
$s-t=a\left(47.2^{49}+\frac{53}{2}\right)$
$\text { Clearly } s > t$
$a_{101}=a_1+100 d=a+2 a .2^{50} 2 a=a\left(2^{51}-1\right)$
$b_{101}=b_1 r^{100}=a \cdot 2^{100}$
Hence $: b_{101} > a_{101}$
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