Tính S = 2+22+222+2222+.......+2222222222

$S=2+22+222+...+ 2 222 222 222 $

$\dfrac{9}{2}S=9+ 99+ 999+....+ 9 999 999 999$

$=10^1-1+10^2-1+10^3-1+...+10^{10}-1$

$=(10^1+10^2+10^3+...+10^{10})-10$

Ta có: $10^1+10^2+...+10^{10}$ là tổng 10 số hạng đầu của CSN $u_1= q=10$

$\Rightarrow \dfrac{9}{2}S=\dfrac{10(1-10^{10})}{1-10}-10$

$=\dfrac{10(10^{10}-1)}{9}-10$

$\Rightarrow S=\dfrac{2}{9}\Big(\dfrac{10(10^{10}-1)}{9}-10\Big)$

$=\dfrac{20(10^{10}-1)}{81}-\dfrac{20}{9}$