Sequences & Series
Series with Repeating Decimals
Grade 11
Question:
<p><strong>Statement-1:</strong> For all natural numbers n, <latex>0.5 + 0.55 + 0.555 + \cdots</latex> (up to n terms) = <latex>\frac{5}{9}\left(n - \frac{1 + 10^{-n}}{9}\right)</latex>.</p><p><strong>Statement-2:</strong> [Not fully legible in source text]</p>
<p>(a) Statement-1 is true, Statement-2 is true; Statement-2 is correct explanation for Statement-1</p>
<p>(b) Statement-1 is true, Statement-2 is true; Statement-2 is not correct explanation for Statement-1</p>
<p>(c) Statement-1 is true, Statement-2 is false</p>
<p>(d) Cannot determine due to incomplete information</p>
Step-by-Step Solution
Key Concept: Decompose repeating decimals and use mathematical induction to prove series sum formulas.
<p>The series <latex>0.5 + 0.55 + 0.555 + \cdots</latex> can be written as <latex>\sum_{k=1}^{n} \frac{5}{9}(1 - 10^{-k})</latex>. The given formula needs verification by mathematical induction. Statement-2 information is incomplete in the source text.</p>
Correct Answer: d