Find the sum to 90 terms of the series $5+55+555+\cdots\cdots$
- $\displaystyle\frac{50}{81}\left[10^{90}-82\right]$
- $\displaystyle\frac{50}{81}\left[10^{90}-83\right]$
- $\displaystyle\frac{50}{81}\left[10^{90}-80\right]$
- $\displaystyle\frac{50}{81}\left[10^{90}-90\right]$
The series is 5(1 + 11 + 111 + ...) = 5/9 * (9 + 99 + 999 + ...) = 5/9 * [(10-1) + (100-1) + ... + (10^90-1)] = 5/9 * [(10 + 100 + ... + 10^90) - 90]. Using geometric sum: 5/9 * [10(10^90 - 1)/9 - 90] = 50/81 * (10^90 - 1) - 50/9 * 90 = 50/81 * (10^90 - 1 - 81) = 50/81 * (10^90 - 82).
The k-th term of the series can be written as 5 multiplied by the sum of a geometric progression with k terms having the first term as 1/10 and common ratio 1/10, giving 5/9(10^k - 1). The sum of 90 such terms becomes 5/9 times the sum of (10^1 + 10^2 + ... + 10^90) minus 90, which simplifies to 5/9 * [10(10^90 - 1)/9 - 90]. Distributing the terms results in 50/81(10^90 - 91) + 50/9(9), yielding the final sum of 50/81[10^90 - 82].