Question

find the sum of the series shown below.
\\(\displaystyle\sum_{k=0}^{4} 3^k\\)
the sum is \\(\square\\).
(simplify your answer.)

Explanation:

Step1: Expand the summation

We need to find the sum \(\sum_{k = 0}^{4}3^{k}\). Expanding this, we get \(3^{0}+3^{1}+3^{2}+3^{3}+3^{4}\).

Step2: Calculate each term

• For \(k = 0\): \(3^{0}=1\)
• For \(k = 1\): \(3^{1}=3\)
• For \(k = 2\): \(3^{2}=9\)
• For \(k = 3\): \(3^{3}=27\)
• For \(k = 4\): \(3^{4}=81\)

Step3: Sum the terms

Now, we add these terms together: \(1 + 3+9 + 27+81\). First, \(1+3 = 4\), then \(4 + 9=13\), then \(13+27 = 40\), and finally \(40+81=121\).

Answer:

\(121\)