# Series in a series

Following series is given upto n terms:

$$1,12,123,1234,12345,123456,1234567,...123456789,1234567890...$$

It's sum up to n terms is defined by the formula:

$$\frac{a(10^{n}-1)-b(n)-c(n)(n+1)}{d}$$

( a,b,c,d are integer constants . )

What is a+b+2c+d?

Hint:

Try splitting each term into a series. $$(12-1)=?$$ And $$(1234-123)=?$$.

Try finding formula upto n terms by summation (then deduce to the above equation) and then compare coefficients to the above equation and just sum to give the answer.

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