How to find a general sum formula for the series: 5+55+555+5555+…..?

I have a question about finding the sum formula of n-th terms.

Here’s the series:

5+55+555+5555+……

What is the general formula to find the sum of n-th terms?

My attempts:

I think I need to separate 5 from this series such that:

5(1+11+111+1111+....)

Then, I think I need to make the statement in the parentheses into a easier sum:

5(1+(10+1)+(100+10+1)+(1000+100+10+1)+.....)

= 5(1n+10(n1)+100(n2)+1000(n3)+....)

Until the last statement, I don’t know how to go further. Is there any ideas to find the general solution from this series?

Thanks

Answer

5+55+555+5555++n fives555
=59(9+99+999+9999++n nines999)
=59(1011+1021+1031++10n1)
=59(101+102+103++10nn)
=59(10n+1109n).

Attribution
Source : Link , Question Author : akusaja , Answer Author : bof

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