Saturday, 12 December 2015

summation - 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).


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