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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