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Find the sum of the first 10 terms of the following sequence. Round to the nearest hundredth if necessary. 27,quad18,quad12,dots Sum of a finite geometric series: S_(n)=(a_(1)-a_(1)r^(n))/(1-r) Answer:

Find the sum of the first 1010 terms of the following sequence. Round to the nearest hundredth if necessary.\newline27,18,12,… 27, \quad 18, \quad 12, \ldots \newlineSum of a finite geometric series:\newlineSn=a1−a1rn1−r S_{n}=\frac{a_{1}-a_{1} r^{n}}{1-r} \newlineAnswer:

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Q. Find the sum of the first 1010 terms of the following sequence. Round to the nearest hundredth if necessary.\newline27,18,12,… 27, \quad 18, \quad 12, \ldots \newlineSum of a finite geometric series:\newlineSn=a1−a1rn1−r S_{n}=\frac{a_{1}-a_{1} r^{n}}{1-r} \newlineAnswer:
  1. Identify sequence type: First, identify the type of sequence given. The sequence 27,18,12,…27, 18, 12, \ldots is a geometric sequence because each term is obtained by multiplying the previous term by a common ratio.
  2. Determine first term: Determine the first term a1a_1 of the sequence. The first term a1a_1 is 2727.
  3. Find common ratio: Find the common ratio rr of the sequence. To find the common ratio, divide the second term by the first term: r=1827=23r = \frac{18}{27} = \frac{2}{3}.
  4. Use sum formula: Use the formula for the sum of the first nn terms of a geometric series: Sn=a1−a1⋅rn1−rS_n = \frac{a_1 - a_1 \cdot r^n}{1 - r}. We need to find the sum of the first 1010 terms, so n=10n = 10.
  5. Plug values into formula: Plug the values into the formula: S10=(27−27×(23)10)(1−23)S_{10} = \frac{(27 - 27 \times (\frac{2}{3})^{10})}{(1 - \frac{2}{3})}.
  6. Calculate sum: Calculate the sum: S10=27−27×(23)1013S_{10} = \frac{27 - 27 \times (\frac{2}{3})^{10}}{\frac{1}{3}}.
  7. Simplify expression: Simplify the expression: S10=27×(1−(23)10)13S_{10} = \frac{27 \times (1 - (\frac{2}{3})^{10})}{\frac{1}{3}}.
  8. Calculate (23)10(\frac{2}{3})^{10}: Calculate (23)10(\frac{2}{3})^{10}: (23)10≈0.01734(\frac{2}{3})^{10} \approx 0.01734 (rounded to five decimal places).
  9. Substitute value into formula: Substitute the value back into the sum formula: S10=27×(1−0.01734)/(1/3)S_{10} = 27 \times (1 - 0.01734) / (1/3).
  10. Perform multiplication and division: Simplify the expression: S10=27×(0.98266)/(13)S_{10} = 27 \times (0.98266) / (\frac{1}{3}).
  11. Calculate final sum: Perform the multiplication and division: S10=27×0.98266×3S_{10} = 27 \times 0.98266 \times 3.
  12. Round to nearest hundredth: Calculate the final sum: S10=79.54794S_{10} = 79.54794.
  13. Round to nearest hundredth: Calculate the final sum: S10=79.54794S_{10} = 79.54794. Round to the nearest hundredth: S10≈79.55S_{10} \approx 79.55.

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