Problem 59
Question
Use the formula for \(S_{n}\) to find the sum of the terms of each geometric sequence. $$\sum_{i=1}^{7} 9(2)^{i}$$
Step-by-Step Solution
Verified Answer
The sum of the given geometric sequence is \(S_7 = 2286\).
1Step 1: Identify the first term, common ratio, and number of terms of the geometric sequence
Based on the given formula \(\sum_{i=1}^{7}9(2)^i\), we can see that the first term of the sequence (a) is \(9 \cdot 2^1 = 18\), the common ratio (r) is 2, and the number of terms (n) is 7.
2Step 2: Apply the formula to find the sum of the geometric sequence
Now that we have identified a = 18, r = 2, and n = 7, we can apply the geometric series sum formula to find the sum (S_n) of the sequence:
\[S_n = a\frac{(1 - r^n)}{(1-r)}\]
3Step 3: Calculate the sum of the terms
By plugging the values a = 18, r = 2, and n = 7 into the formula, we get:
\[S_7 = 18\frac{(1 - 2^7)}{(1 - 2)}\]
Simplify the expression inside the parentheses:
\[S_7 = 18\frac{(1 - 128)}{(-1)}\]
Subtract the values inside the parentheses:
\[S_7 = 18\frac{(-127)}{(-1)}\]
Now, divide -127 by -1 and multiply the result by 18:
\[S_7 = 18(127)\]
\[S_7 = 2286\]
4Step 4: State the sum of the geometric sequence
The sum of the first 7 terms of the geometric sequence \(\sum_{i=1}^{7}9(2)^i\) is 2286.
Key Concepts
Geometric Series Sum FormulaCommon RatioFirst Term
Geometric Series Sum Formula
A geometric series is a series of terms that have a constant ratio between successive terms. The sum of a finite geometric series can be calculated using a specific formula, which is vital for evaluating such series efficiently.
In a typical geometric series where the first term is denoted by \(a\), the common ratio by \(r\), and the number of terms by \(n\), the formula for finding the sum \(S_n\) is:
In the given exercise, the first term \(a\) is found to be 18, the common ratio \(r\) is 2, and the number of terms \(n\) is 7. By substituting these values into the formula, we can calculate the sum of the series efficiently.
Understanding the structure of this formula helps in seeing how each part interacts—the numerator \((1 - r^n)\) handles the exponential growth, while the denominator \((1-r)\) normalizes this growth along the series.
In a typical geometric series where the first term is denoted by \(a\), the common ratio by \(r\), and the number of terms by \(n\), the formula for finding the sum \(S_n\) is:
- \( S_n = a\frac{(1 - r^n)}{(1-r)} \) when \(r eq 1\)
In the given exercise, the first term \(a\) is found to be 18, the common ratio \(r\) is 2, and the number of terms \(n\) is 7. By substituting these values into the formula, we can calculate the sum of the series efficiently.
Understanding the structure of this formula helps in seeing how each part interacts—the numerator \((1 - r^n)\) handles the exponential growth, while the denominator \((1-r)\) normalizes this growth along the series.
Common Ratio
The common ratio is a critical part of understanding a geometric sequence. It's the factor by which each consecutive term is multiplied to get the next term. In mathematical terms, it is often represented as \(r\).
To identify the common ratio in a series, look at the fraction formed by dividing any term by its preceding term: \( r = \frac{a_{2}}{a_{1}} \). This ratio remains constant throughout the series.
In the exercise provided, the common ratio is given as \(2\). This means each term is double the previous one. Knowing the common ratio allows us to predict the next term in the series simply by multiplying the current term by the common ratio entirely.
This consistency in growth is what gives geometric sequences their use in modeling exponential growth phenomena such as population growth, radioactive decay, and investment growth over time.
To identify the common ratio in a series, look at the fraction formed by dividing any term by its preceding term: \( r = \frac{a_{2}}{a_{1}} \). This ratio remains constant throughout the series.
In the exercise provided, the common ratio is given as \(2\). This means each term is double the previous one. Knowing the common ratio allows us to predict the next term in the series simply by multiplying the current term by the common ratio entirely.
This consistency in growth is what gives geometric sequences their use in modeling exponential growth phenomena such as population growth, radioactive decay, and investment growth over time.
First Term
The first term of a geometric sequence is the starting point from which the series begins. It is represented by \(a\) in the formulas.
Determining the first term is crucial because it sets the entire series' base value. Without it, we cannot accurately determine the growth pattern or compute the sum of the series using the geometric series sum formula.
In the problem given, the first term \(a\) was calculated to be 18 by evaluating the expression \(9 \cdot 2^1\). This initial term informs us of the point from which each subsequent term will be calculated using the common ratio.
Understanding the role of the first term helps in placing the sequence within a broader context and also aids in visually modeling the growth pattern of the series either on a conceptual level or using graphical tools.
Determining the first term is crucial because it sets the entire series' base value. Without it, we cannot accurately determine the growth pattern or compute the sum of the series using the geometric series sum formula.
In the problem given, the first term \(a\) was calculated to be 18 by evaluating the expression \(9 \cdot 2^1\). This initial term informs us of the point from which each subsequent term will be calculated using the common ratio.
Understanding the role of the first term helps in placing the sequence within a broader context and also aids in visually modeling the growth pattern of the series either on a conceptual level or using graphical tools.
Other exercises in this chapter
Problem 58
Find \(S_{8}\) for each arithmetic sequence described below. $$a_{n}=4 n+1$$
View solution Problem 59
Find the indicated term of each binomial expansion. $$\left(2 y^{2}+z\right)^{10} ; \text { eighth term }$$
View solution Problem 59
Write each series using summation notation. -1+2-3+4-5+6-7
View solution Problem 59
Find \(S_{8}\) for each arithmetic sequence described below. $$a_{n}=3 n+4$$
View solution