Step 1: Understanding the Concept:
In an $S_N1$ reaction, the rate depends on the stability of the carbocation formed after the leaving group departs.
Greater the stability of the carbocation, greater will be the reactivity.
Carbocation stability order:
\[
3^\circ>2^\circ>1^\circ
\]
Step 2: Classification of Given Compounds:
A. $(CH_3)_3CBr$
Forms a tertiary ($3^\circ$) carbocation → Most stable.
B. $CH_3CH_2CH_2CH_2Br$
Forms a primary ($1^\circ$) carbocation → Least stable.
C. $(CH_3)_2CHCH_2Br$
Also forms a primary ($1^\circ$) carbocation → Slightly more stable than straight chain due to inductive effect, but still primary.
D. $CH_3CH(Br)CH_2CH_3$
Forms a secondary ($2^\circ$) carbocation → More stable than primary.
Thus, increasing order of reactivity in $S_N1$ reaction:
\[
1^\circ<2^\circ<3^\circ
\]
So,
\[
B<C<D<A
\]
Step 3: Final Answer:
The correct increasing order of reactivity in $S_N1$ reaction is $B<C<D<A$.