Discussion
Find the sum of the series: \(5 + 55 + 555 + ...\) up to n terms.
Answer: A
The sum S can be written as \(S = 5(1+11+111+...)\). Multiply and divide by 9: \(S = \frac{5}{9}(9+99+999+...)\). This can be written as \(S = \frac{5}{9}[(10-1)+(100-1)+(1000-1)+...]\). \(S = \frac{5}{9}[(10+100+1000+...) - (1+1+...+n \text{ times})]\). The first part is a G.P. Its sum is \(10\frac{10^n-1}{10-1} = \frac{10}{9}(10^n-1)\). So \(S = \frac{5}{9}[\frac{10}{9}(10^n-1) - n] = \frac{5}{81}[10(10^n-1)-9n] = \frac{5}{81}(10^{n+1}-10-9n)\).
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