Sequences & Series
Sum of Series
Grade 11
Question:
<p>The sum of first 20 terms of the sequence 0.7, 0.77, 0.777, ... is</p>
<p>\(\frac{7}{81}(179 - 10^{-20})\)</p>
<p>\(\frac{7}{9}(99 - 10^{-20})\)</p>
<p>\(\frac{7}{81}(179 + 10^{-20})\)</p>
<p>\(\frac{7}{9}(99 + 10^{-20})\)</p>
Step-by-Step Solution
Key Concept: Recognize that each term can be written as 7/9 × (1 - 10^(-n)), converting the repeating decimal pattern into a geometric series that can be summed using standard formulas.
<p><strong>Step 1:</strong> Express each term as a fraction.</p><p>0.7 = 7/9 × (1 - 1/10)</p><p>0.77 = 7/9 × (1 - 1/100)</p><p>0.777 = 7/9 × (1 - 1/1000)</p><p>In general, the nth term: aₙ = 7/9 × (1 - 10^(-n))</p><p><strong>Step 2:</strong> Find S₂₀ = Σ(n=1 to 20) [7/9 × (1 - 10^(-n))]</p><p>S₂₀ = (7/9) × Σ(n=1 to 20) [1 - 10^(-n)]</p><p>S₂₀ = (7/9) × [20 - Σ(n=1 to 20) 10^(-n)]</p><p><strong>Step 3:</strong> Calculate the geometric series Σ(n=1 to 20) 10^(-n)</p><p>This is a geometric series with first term a = 1/10, ratio r = 1/10, and 20 terms.</p><p>Sum = (1/10) × [(1 - (1/10)²⁰)/(1 - 1/10)] = (1/10) × [(1 - 10^(-20))/(9/10)]</p><p>Sum = (1/9) × (1 - 10^(-20))</p><p><strong>Step 4:</strong> Substitute back:</p><p>S₂₀ = (7/9) × [20 - (1/9) × (1 - 10^(-20))]</p><p>S₂₀ = (7/9) × [(180 - 1 + 10^(-20))/9]</p><p>S₂₀ = (7/9) × [(179 + 10^(-20))/9]</p><p>S₂₀ = 7(179 + 10^(-20))/81 = <strong>(1253 + 7×10^(-20))/81</strong> or approximately <strong>1253/81</strong></p>
Correct Answer: A