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Find the sum of the finite geometric series. \newline∑n=162⋅2(n–1)\sum_{n = 1}^{6} 2 \cdot 2^{(n – 1)}\newline______

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Q. Find the sum of the finite geometric series. \newline∑n=162⋅2(n–1)\sum_{n = 1}^{6} 2 \cdot 2^{(n – 1)}\newline______
  1. Identify terms and ratio: Identify the first term and the common ratio of the geometric series.\newlineThe first term a1a_1 is given when n=1n = 1, which is 2×21–1=2×20=2×1=22 \times 2^{1 – 1} = 2 \times 2^0 = 2 \times 1 = 2.\newlineThe common ratio rr is 22 because each term is multiplied by 22 to get the next term.
  2. Use series formula: Use the formula for the sum of a finite geometric series.\newlineThe sum SnS_n of the first nn terms of a geometric series is given by the formula Sn=a1×(1−rn)/(1−r)S_n = a_1 \times (1 - r^n) / (1 - r), where a1a_1 is the first term, rr is the common ratio, and nn is the number of terms.
  3. Plug in values: Plug the values into the formula.\newlineHere, a1=2a_1 = 2, r=2r = 2, and n=6n = 6.\newlineS6=2×(1−26)/(1−2)S_6 = 2 \times (1 - 2^6) / (1 - 2)
  4. Calculate sum: Calculate the sum.\newlineS6=2×(1−64)/(1−2)S_6 = 2 \times (1 - 64) / (1 - 2)\newlineS6=2×(−63)/(−1)S_6 = 2 \times (-63) / (-1)\newlineS6=2×63S_6 = 2 \times 63\newlineS6=126S_6 = 126

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