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Evaluate the geometric series for the given number of terms:

54 + 18 + 6 + 2 + ...; n = 100

A. [tex]\frac{54}{3}[/tex]

B. [tex]\frac{54}{2}[/tex]

C. [tex]54\left(\frac{2}{3}\right)^{100}[/tex]

D. [tex]54\left(\frac{1}{3}\right)^{100}[/tex]

Answer :

Final answer:

To evaluate the geometric series 54 + 18 + 6 + 2 + ... for 100 terms, we use the formula Sn = a1 * (1 - rn) / (1 - r), where a1 is the first term, r is the common ratio, and n is the number of terms. After calculation, the sum S100 ≈ 54 * (3/2) = 81, which does not match the given options.

Explanation:

To evaluate the geometric series for the given number of terms, we can use the formula for the sum of a finite geometric series:


Sn = a1 * (1 - rn) / (1 - r), where

  • a1 is the first term of the series,
  • r is the common ratio between the terms,
  • n is the number of terms.

Given the series 54 + 18 + 6 + 2 + ..., we can see:

  • a1 = 54,
  • The common ratio r = 18/54 = 1/3,
  • n = 100 terms.

Plugging these values into the formula:
S100 = 54 * (1 - (1/3)100) / (1 - 1/3) = 54 * (1 - (1/3)100) / (2/3)

This simplifies to:
S100 = 54 * (3/2) * (1 - (1/3)100)

Since (1/3)100 is an exceedingly small number, it approaches 0 when considering the first 100 terms of the series. Therefore, the sum of the 100 terms approximately equals:

S100 ≈ 54 * (3/2) = 81.

However, this does not match with any of the provided options, so it seems there might be an error in the options.

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