Question
For the infinite series 162 + 54 + 18 + 6 + …
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 algebraic work.
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 algebraic work.
Answers
GPT 3.5
e) The series can be represented in summation notation as:
\[\sum_{n=1}^{\infty} \frac{162}{3^{n-1}}\]
f) To find the sum of this infinite series, we can use the formula for the sum of an infinite geometric series:
\[S = \frac{a}{1 - r}\]
Where:
a = first term = 162
r = common ratio = 1/3
So, plugging in the values:
\[S = \frac{162}{1 - 1/3}\]
\[S = \frac{162}{2/3}\]
\[S = 162 \times \frac{3}{2}\]
\[S = 243\]
Therefore, the sum of the infinite series 162 + 54 + 18 + 6 + … is 243.
\[\sum_{n=1}^{\infty} \frac{162}{3^{n-1}}\]
f) To find the sum of this infinite series, we can use the formula for the sum of an infinite geometric series:
\[S = \frac{a}{1 - r}\]
Where:
a = first term = 162
r = common ratio = 1/3
So, plugging in the values:
\[S = \frac{162}{1 - 1/3}\]
\[S = \frac{162}{2/3}\]
\[S = 162 \times \frac{3}{2}\]
\[S = 243\]
Therefore, the sum of the infinite series 162 + 54 + 18 + 6 + … is 243.