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Find the sum of the infinite geometric series.\newline1+12+14+18+…1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots\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.\newline1+12+14+18+…1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots\newlineWrite your answer as an integer or a fraction in simplest form.\newline__\_\_
  1. Calculate Rolls Needed: To solve this problem, we need to determine how many rolls of tape the electrician needs to order to have 8,0008,000 centimeters of electrical tape, given that each roll contains 2,0002,000 centimeters of tape. We will use division to calculate the number of rolls required.
  2. Division Calculation: We divide the total amount of tape needed by the amount of tape on each roll to find the number of rolls needed.\newlineCalculation: 8,000 cm÷2,000 cm/roll=4 rolls8,000 \, \text{cm} \div 2,000 \, \text{cm/roll} = 4 \, \text{rolls}
  3. Infinite Geometric Series: The given series is an infinite geometric series with the first term a=1a = 1 and the common ratio r=12r = \frac{1}{2}. To find the sum of an infinite geometric series, we use the formula S=a(1−r)S = \frac{a}{(1 - r)}, where SS is the sum, aa is the first term, and rr is the common ratio. This formula is only valid if the absolute value of rr is less than 11, which is true in this case since ∣12∣<1|\frac{1}{2}| < 1.
  4. Substitute Values: We substitute the values of aa and rr into the formula to find the sum of the series.\newlineCalculation: S=1(1−12)=1(12)=1×(21)=2S = \frac{1}{(1 - \frac{1}{2})} = \frac{1}{(\frac{1}{2})} = 1 \times (\frac{2}{1}) = 2

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