Problem 19
Question
Evaluate the finite series for the specified number of terms. $$ 1+2+4+\ldots ; n=8 $$
Step-by-Step Solution
Verified Answer
The sum of the first 8 terms of the series is 255.
1Step 1: Identify the values
Identify the first term (a), the common ratio (r), and the number of terms (n). In this series, the first term a is 1, the common ratio r is 2 (since each term is doubled), and the number of terms n is given as 8.
2Step 2: Plug the values into the formula
Substitute these values into the formula for the sum of a geometric series: \( S_n = a*(1 - r^n)/(1 - r) = 1*(1 - 2^8)/(1 - 2) \)
3Step 3: Calculate the sum
Calculate the sum by simplifying the expression. Since 2^8 is 256 and (1 - 256) gives -255, the equation becomes: \( S_n = 1*(-255)/-1 = 255 \)
Other exercises in this chapter
Problem 18
Write an explicit formula for each sequence. Then find \(a_{12}\) $$ 4,5,6,7,8, \dots $$
View solution Problem 19
Find the area under each curve for the domain \(0 \leq x \leq 1\) $$ f(x)=x+2 $$
View solution Problem 19
For each sum, find the number of terms, the first term, and the last term. Then evaluate the series. $$ \sum_{n=1}^{5}(2 n-1) $$
View solution Problem 19
Write the explicit formula for each sequence. Then generate the first five terms. $$ a_{1}=1024, r=0.5 $$
View solution