Mathematics Olympiads: A curious calculation and its cute proof !!

Explain why the following calculations hold:

$1.9 + 2 =11$
$12.9 + 3 = 111$
$123.9 + 4 = 1111$
$1234.9 + 5 = 11111$
$12345.9 + 6 = 111111$
$123456.9 + 7 = 1111111$
$1234567.9 + 8 = 11111111$
$12345678.9 + 9 = 111111111$
$123456789.9 + 10 = 11111 11111$

Hint:

Show that $(10^{n-1}+2.10^{n-2}+3.10^{n-3}+ \ldots + n)(10-1) + (n+1)=\frac{10^{n+1}-1}{9}$

More later,
Nalin Pithwa

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