What is the sum of the series $0.5 + 0.55 + 0.555 + ....$ to $n$ terms?
- $\dfrac {5}{9} \left [n - \dfrac {2}{9} \left (1 - \dfrac {1}{10^{n}}\right )\right ]$
- $\dfrac {1}{9} \left [5 - \dfrac {2}{9} \left (1 - \dfrac {1}{10^{n}}\right )\right ]$
- $\dfrac {1}{9} \left [n - \dfrac {5}{9} \left (1 - \dfrac {1}{10^{n}}\right )\right ]$
- $\dfrac {5}{9} \left [n - \dfrac {1}{9} \left (1 - \dfrac {1}{10^{n}}\right )\right ]$
Reveal answer
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Correct answer
Explanation
The series is 0.5 + 0.55 + 0.555... = 5/9 * (0.9 + 0.99 + 0.999...) = 5/9 * [(1-0.1) + (1-0.01) + (1-0.001)...] = 5/9 * [n - (0.1 + 0.01 + ...)] = 5/9 * [n - 1/9 * (1 - 1/10^n)].
AI explanation
Rewrite the series as 5/10 + 55/100 + 555/1000 + ... to n terms. Factor out 5/9 to get 5/9 * (9/10 + 99/100 + 999/1000 + ...). The inner series is the sum of (1 - 1/10^k) from k=1 to n, which separates into the sum of 1 (equaling n) minus the sum of (1/10)^k. The geometric sum of (1/10)^k to n terms is (1/10)(1 - 1/10^n) / (1 - 1/10) = 1/9 * (1 - 1/10^n). The result is 5/9 * [n - 1/9 * (1 - 1/10^n)].