$\text{7+7²+7³+..............+ $7^{100}$ }$
$\text{= (7+7²+7³+$7^{4}$) +($7^{5}$+$7^{6}$ +$7^{7}$ +$7^{8}$ )+($7^{97}$ +$7^{98}$ +$7^{99}$ +$7^{100}$ )}$
$\text{=7.(1+7+7²+7³)+$7^{5}$ .(1+7+7²+7³)+...........+ $7^{97}$(1+7+7²+7³)}$
$\text{=(1+7+7²+7³).(7+$7^{5}$ +..............+$7^{97}$)}$
$\text{=400.(7+$7^{5}$ +..............+$7^{97}$)}$
$\text{=400.7.(1+$7^{4}$+..............+$7^{96}$)}$
$\text{=2800.(1+$7^{4}$+..............+$7^{96}$) chia hết cho 700 (dpcm)}$