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Find the sum of the infinite geometric series.\newline8+2+12+18+8 + 2 + \frac{1}{2} + \frac{1}{8} + \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.\newline8+2+12+18+8 + 2 + \frac{1}{2} + \frac{1}{8} + \newlineWrite your answer as an integer or a fraction in simplest form.\newline__\_\_
  1. Identify first term and ratio: Identify the first term a1a_1 and the common ratio rr of the geometric series.\newlineThe first term a1a_1 is the first number in the series, which is 88.\newlineTo find the common ratio rr, we divide the second term by the first term.\newliner=a2a1=28=14r = \frac{a_2}{a_1} = \frac{2}{8} = \frac{1}{4}
  2. Use sum formula: Use the formula for the sum of an infinite geometric series, which is S=a11−rS = \frac{a_1}{1 - r}, where SS is the sum, a1a_1 is the first term, and rr is the common ratio.\newlineHere, a1=8a_1 = 8 and r=14r = \frac{1}{4}.\newlineNow, plug these values into the formula to find the sum.\newlineS=81−14S = \frac{8}{1 - \frac{1}{4}}
  3. Simplify expression: Simplify the expression to find the sum.\newlineS=834S = \frac{8}{\frac{3}{4}}\newlineTo divide by a fraction, multiply by its reciprocal.\newlineS=8×43S = 8 \times \frac{4}{3}\newlineS=323S = \frac{32}{3}

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