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Home » Find the sum of the series \( 1 + 3 + 5 + \dots + 99 \). - Portal da AcÃºstica

Find the sum of the series \( 1 + 3 + 5 + \dots + 99 \). - Portal da AcÃºstica

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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\frac{1 + \frac{\sqrt{7}}{3}}{1 - \sqrt{7}} = \frac{\frac{3 + \sqrt{7}...
Numerator: $ (3 + \sqrt{7})(1 + \sqrt{7}) = 3 + 3\sqrt{7} + \sqrt{7} +...
Denominator: $ 3(1 - 7) = 3(-6) = -18 $.
So ratio: $ \frac{10 + 4\sqrt{7}}{-18} = -\frac{5 + 2\sqrt{7}}{9} $.
Now $ x = -\frac{\sqrt{7}}{3} $:
\frac{1 - \frac{\sqrt{7}}{3}}{1 + \sqrt{7}} = \frac{\frac{3 - \sqrt{7}...
Multiply by $ 1 - \sqrt{7} $:
Numerator: $ (3 - \sqrt{7})(1 - \sqrt{7}) = 3 - 3\sqrt{7} - \sqrt{7} +...
Denominator: $ 3(1 - 7) = -18 $.
Ratio: $ \frac{10 - 4\sqrt{7}}{-18} = -\frac{5 - 2\sqrt{7}}{9} $.

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