$2^{3\mathrm{n}}-7\mathrm{n}-1$ s divisible by 

Let  $\mathrm{P}\left(\mathrm{n}\right)=2^{3\mathrm{n}}-7\mathrm{n}-1$

$\Rightarrow \mathrm{P}\left(1\right)=0,\mathrm{P}\left(2\right)=49$

P(1) and P(2) are divisible by 49.

$\begin{array}{c}\text{ Let }\mathrm{P}\left(\mathrm{k}\right)=2^{3\mathrm{k}}-7\mathrm{k}-1=49\mathrm{l}\\ \mathrm{P}(\mathrm{k}+1)=2^{3\mathrm{k}+3}-7\mathrm{k}-8\\ =8\left(49\right)+7\mathrm{k}+1)-7\mathrm{k}-8\\ =49(8/)+49\mathrm{k}=49\mathrm{\lambda }\end{array}$

where, $\mathrm{\lambda }=8\mathrm{I}+\mathrm{k}$ which is an integer.