Problem 18
Question
Evaluate the finite series for the specified number of terms. $$ 4+12+36+\ldots ; n=6 $$
Step-by-Step Solution
Verified Answer
The sum of the first 6 terms of the series is 1460.
1Step 1: Identify values for a, r, and n
Our series begins with an initial term (a) of 4. Each successive term is obtained by multiplying the previous one by the common ratio (r), which is 3. We need the sum of the first 6 terms (n). So, we have \(a = 4\), \(r = 3\), and \(n = 6\).
2Step 2: Plug values into sum formula
Let's insert \(a = 4\), \(r = 3\), and \(n = 6\) into the formula \(S_n = a \frac{{1-r^n}}{{1-r}}\). Substituting, we get \(S_6 = 4 \frac{{1-3^6}}{{1-3}}\).
3Step 3: Perform the calculations
After calculations are done, \(S_6 = 4 \frac{{1-729}}{{-2}} = 4 \cdot 365 = 1460\).
Other exercises in this chapter
Problem 17
Write a recursive formula for each sequence. Then find the next term. $$ \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}, \dots $$
View solution Problem 18
Find the area under each curve for the domain \(0 \leq x \leq 1\) $$ y=-x^{2}+2 $$
View solution Problem 18
Use summation notation to write each arithmetic series for the specified number of terms. $$ (-3)+(-6)+(-9)+\ldots ; n=5 $$
View solution Problem 18
Write the explicit formula for each sequence. Then generate the first five terms. $$ a_{1}=7, r=1 $$
View solution