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Find the sum of the first 8 terms in the following geometric series. Do not round your answer. 2+8+32+dots

Find the sum of the first 88 terms in the following geometric series. Do not round your answer.\newline2+8+32+… 2+8+32+\ldots

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Full solution

Q. Find the sum of the first 88 terms in the following geometric series. Do not round your answer.\newline2+8+32+… 2+8+32+\ldots
  1. Identifying the first term and common ratio: To find the sum of the first 88 terms of a geometric series, we need to identify the first term (aa) and the common ratio (rr) of the series. The first term is given as 22. To find the common ratio, we divide the second term by the first term.\newlineCalculation: r=82=4r = \frac{8}{2} = 4
  2. Using the formula for the sum of the first n terms: Now that we have the first term (a = 22) and the common ratio (r = 44), we can use the formula for the sum of the first n terms of a geometric series: S_n = \frac{a(11 - r^n)}{11 - r}, where n is the number of terms.\newlineWe want to find the sum of the first 88 terms, so n = 88.
  3. Substituting values into the formula: Plugging the values into the formula, we get S8=2(1−48)1−4S_8 = \frac{2(1 - 4^8)}{1 - 4}.\newlineCalculation: S8=2(1−48)1−4S_8 = \frac{2(1 - 4^8)}{1 - 4}
  4. Calculating 484^8: Before we continue, let's calculate 484^8 to avoid any math errors later on.\newlineCalculation: 48=655364^8 = 65536
  5. Substituting 484^8 into the formula: Now we substitute 484^8 into the formula.\newlineCalculation: S8=2(1−65536)1−4S_8 = \frac{2(1 - 65536)}{1 - 4}
  6. Simplifying the expression: Simplify the expression inside the parentheses.\newlineCalculation: S8=2(−65535)−3S_8 = \frac{2(-65535)}{-3}
  7. Calculating the sum of the first 88 terms: Now we divide −65535-65535 by −3-3 and multiply by 22 to find the sum of the first 88 terms.\newlineCalculation: S8=2×655353=2×21845=43690S_8 = 2 \times \frac{65535}{3} = 2 \times 21845 = 43690

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