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Find the sum of the infinite geometric series.\newline−5−1−15−125+-5 - 1 - \frac{1}{5} - \frac{1}{25} + \newlineWrite your answer as an integer or a fraction in simplest form.\newline______

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Q. Find the sum of the infinite geometric series.\newline−5−1−15−125+-5 - 1 - \frac{1}{5} - \frac{1}{25} + \newlineWrite your answer as an integer or a fraction in simplest form.\newline______
  1. Identify Terms and Ratio: To find the sum of an infinite geometric series, we need to identify the first term aa and the common ratio rr of the series. The formula for the sum of an infinite geometric series is S=a1−rS = \frac{a}{1 - r}, where ∣r∣<1|r| < 1. In the given series, the first term is −5-5 and each subsequent term is 15\frac{1}{5} times the previous term, so the common ratio is −15-\frac{1}{5}.
  2. Apply Sum Formula: We can now apply the formula for the sum of an infinite geometric series: S=a1−rS = \frac{a}{1 - r}. Substituting the values we have, S=−51−(−15)S = \frac{-5}{1 - (-\frac{1}{5})}.
  3. Substitute Values: Now we perform the calculation: S=(−5)/(1+1/5)S = (-5) / (1 + 1/5).\newlineFirst, we convert 11 to a fraction with a denominator of 55 to combine it with 1/51/5: 1=5/51 = 5/5.\newlineSo, S=(−5)/(5/5+1/5)S = (-5) / (5/5 + 1/5).
  4. Perform Calculation: Next, we add the fractions in the denominator: (55+15)=65(\frac{5}{5} + \frac{1}{5}) = \frac{6}{5}. So, S=−565S = \frac{-5}{\frac{6}{5}}.
  5. Combine Fractions: To divide by a fraction, we multiply by its reciprocal. Therefore, S=(−5)×(56)S = (-5) \times \left(\frac{5}{6}\right).
  6. Multiply by Reciprocal: Now we multiply the numerators and the denominators: S=−5×56S = \frac{-5 \times 5}{6}.
  7. Final Multiplication: The multiplication gives us: S=−256S = -\frac{25}{6}. This is the sum of the infinite geometric series in simplest fractional form.

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