For the infinite series 162 + 54 + 18 + 6 + … a) Is the series arithmetic, geometric, or neither? b) If it’s arithmetic, identify the common difference. If it’s geometric, identify the common ratio. c) What are the next three terms of this series? d) Does the series converge or diverge? Explain how you know. e) Write the series in summation notation where the lower limit is n=1. f) What is the sum of this infinite series? Reminder: You will be graded for showing the correct process as well as the correct answer using handwritten algebraic work.
a) This series is geometric.
b) The common ratio can be found by dividing any term by the term before it. For example, 54/162 = 1/3. Therefore, the common ratio is 1/3.
c) The next three terms of the series are 2, 2/3, and 2/9.
d) The series converges because the common ratio is between -1 and 1. As the common ratio is less than 1, the series converges.
e) The series in summation notation starting at n=1 is:
\[\sum_{n=1}^{\infty} 162 \times \left(\frac{1}{3}\right)^{n-1}\]
f) The sum of this infinite series can be calculated using the formula for the sum of an infinite geometric series:
\[S = \frac{a}{1-r}\]
where a is the first term and r is the common ratio.
In this case, a = 162 and r = 1/3.
Therefore, the sum of the infinite series is:
\[S = \frac{162}{1 - 1/3} = \frac{162}{2/3} = 243\]