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Find the sum of the first 9 terms of the following series, to the nearest integer. 54,36,24,dots Answer:

Find the sum of the first 99 terms of the following series, to the nearest integer.\newline54,36,24,… 54,36,24, \ldots \newlineAnswer:

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Full solution

Q. Find the sum of the first 99 terms of the following series, to the nearest integer.\newline54,36,24,… 54,36,24, \ldots \newlineAnswer:
  1. Identify pattern: Identify the pattern in the series.\newlineThe series is 54,36,24,…54, 36, 24, \ldots which seems to be a geometric series because each term is multiplied by a common ratio to get the next term. To find the common ratio (r)(r), we divide the second term by the first term.\newliner=3654r = \frac{36}{54}
  2. Calculate common ratio: Calculate the common ratio.\newliner=3654=23r = \frac{36}{54} = \frac{2}{3}\newlineThe common ratio is 23\frac{2}{3}.
  3. Use formula for sum: Use the formula for the sum of the first nn terms of a geometric series.\newlineThe formula for the sum of the first nn terms (SnS_n) of a geometric series is:\newlineSn=a×(1−rn)/(1−r)S_n = a \times (1 - r^n) / (1 - r), where aa is the first term and rr is the common ratio.\newlineFor this series, a=54a = 54, r=2/3r = 2/3, and n=9n = 9.
  4. Substitute values and calculate: Substitute the values into the formula and calculate the sum.\newlineS9=54×(1−(23)9)/(1−23)S_9 = 54 \times (1 - (\frac{2}{3})^9) / (1 - \frac{2}{3})
  5. Evaluate expression: Evaluate the expression.\newlineS9=54×(1−(23)9)/(13)S_9 = 54 \times (1 - (\frac{2}{3})^9) / (\frac{1}{3})\newlineFirst, calculate (23)9(\frac{2}{3})^9.\newline(23)9=(2939)(\frac{2}{3})^9 = (\frac{2^9}{3^9})
  6. Calculate powers: Calculate (29)(2^9) and (39)(3^9).29=5122^9 = 51239=196833^9 = 19683Now, substitute these values back into the expression.
  7. Continue calculation: Continue the calculation.\newlineS9=54×(1−51219683)/(13)S_9 = 54 \times (1 - \frac{512}{19683}) / (\frac{1}{3})\newlineS9=54×(1968319683−51219683)/(13)S_9 = 54 \times (\frac{19683}{19683} - \frac{512}{19683}) / (\frac{1}{3})\newlineS9=54×(1917119683)/(13)S_9 = 54 \times (\frac{19171}{19683}) / (\frac{1}{3})
  8. Simplify expression: Simplify the expression.\newlineS9=54×(19171/19683)×3S_9 = 54 \times (19171/19683) \times 3\newlineS9=54×3×(19171/19683)S_9 = 54 \times 3 \times (19171/19683)\newlineS9=162×(19171/19683)S_9 = 162 \times (19171/19683)
  9. Perform division: Perform the division.\newlineS9=162×(19171/19683)S_9 = 162 \times (19171/19683)\newlineS9≈162×0.974S_9 \approx 162 \times 0.974
  10. Multiply for sum: Multiply to find the sum.\newlineS9≈162×0.974S_9 \approx 162 \times 0.974\newlineS9≈157.788S_9 \approx 157.788
  11. Round to nearest integer: Round to the nearest integer. S9≈158S_9 \approx 158

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