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Find the sum of the first 7 terms of the following series, to the nearest integer. 6,(9)/(2),(27)/(8),dots Answer:

Find the sum of the first 77 terms of the following series, to the nearest integer.\newline6,92,278,… 6, \frac{9}{2}, \frac{27}{8}, \ldots \newlineAnswer:

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Full solution

Q. Find the sum of the first 77 terms of the following series, to the nearest integer.\newline6,92,278,… 6, \frac{9}{2}, \frac{27}{8}, \ldots \newlineAnswer:
  1. Identify pattern: Identify the pattern of the series. The series starts with 66 and each subsequent term is multiplied by (3/2)(3/2) to get the next term. This is a geometric series with the first term a=6a = 6 and the common ratio r=(3/2)r = (3/2).
  2. Use formula for sum: Use the formula for the sum of the first nn terms of a geometric series, which is Sn=a(1−rn)/(1−r)S_n = a(1 - r^n) / (1 - r), where SnS_n is the sum of the first nn terms, aa is the first term, rr is the common ratio, and nn is the number of terms.
  3. Plug in values: Plug in the values for the first 77 terms: a=6a = 6, r=32r = \frac{3}{2}, and n=7n = 7. Calculate the sum S7=6(1−(32)7)/(1−32)S_7 = 6\left(1 - \left(\frac{3}{2}\right)^7\right) / \left(1 - \frac{3}{2}\right).
  4. Calculate (32)7(\frac{3}{2})^7: Calculate (32)7(\frac{3}{2})^7. This is 3727=2187128\frac{3^7}{2^7} = \frac{2187}{128}.
  5. Substitute in formula: Substitute the value of (32)7(\frac{3}{2})^7 into the sum formula: S7=6(1−2187128)/(1−32)S_7 = 6(1 - \frac{2187}{128}) / (1 - \frac{3}{2}).
  6. Simplify expression: Simplify the expression: S7=6(1−2187128)(−12).S_7 = \frac{6(1 - \frac{2187}{128})}{(-\frac{1}{2})}.
  7. Simplify numerator: Simplify the numerator: 1−2187128=(128128)−(2187128)=128−2187128=−2059128.1 - \frac{2187}{128} = \left(\frac{128}{128}\right) - \left(\frac{2187}{128}\right) = \frac{128 - 2187}{128} = -\frac{2059}{128}.
  8. Substitute numerator: Substitute the simplified numerator into the sum formula: S7=6(−2059/128)(−1/2)S_7 = \frac{6(-2059/128)}{(-1/2)}.
  9. Simplify denominator: Simplify the denominator: −12-\frac{1}{2} is the same as multiplying by −2-2. So, S7=6(−2059128)×−2S_7 = 6(-\frac{2059}{128}) \times -2.
  10. Multiply terms: Multiply the terms: S7=−12×(−2059/128)S_7 = -12 \times (-2059/128).
  11. Simplify multiplication: Simplify the multiplication: S7=24708128S_7 = \frac{24708}{128}.
  12. Divide for sum: Divide to find the sum to the nearest integer: S7≈24708128≈193S_7 \approx \frac{24708}{128} \approx 193.

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