An Unusual Development

It can be proved that any real number $x\in ]0,1]$ can be written uniquely in the form:
$$x=\frac{1}{a_1}-\frac{1}{a_1\times a_2}+\frac{1}{a_1\times a_2\times a_3}-\dots$$ $a_i$ being a sequence of strictly increasing integers.

For example, $e-2=\frac{1}{1}-\frac{1}{1\times 3}+\frac{1}{1\times 3\times 6}-\frac{1}{1\times 3\times 6\times 14}+\frac{1}{1\times 3\times 6\times 14\times 142}-\dots$

What is $a_{100}$ when developping the Champernowne constant $C=0.123456789101112\dots$
Give your answer in compressed form: 10 first digits [ Omitted digits count ] 10 last digits

Example: 1701411834[19]5884105727 for $M_{127}=2^{127}-1$

[My timing : 7 sec]