Problem 12
Question
Find the sum of each series. $$\sum_{i=1}^{5}(-2)^{-i}$$
Step-by-Step Solution
Verified Answer
The sum of the series is \(-\frac{11}{32}\).
1Step 1: Identify the Series
The given series is \ \ \( \sum_{i=1}^{5}(-2)^{-i} \). This is a finite geometric series with the first term being \( a = (-2)^{-1} \) and the common ratio being \( r = (-2)^{-1} \).
2Step 2: Calculate the First Term
To get the first term (\(a\)), we substitute \(i = 1\) into the series expression: \((-2)^{-1} = -\frac{1}{2} \).
3Step 3: Find the Common Ratio
The common ratio \(r\) is \(-2^{-1}\) since each subsequent term is multiplied by \((-2)^{-1} \approx -\frac{1}{2} \).
4Step 4: Use the Formula for Sum of Finite Geometric Series
The sum of a finite geometric series \(S_n\) can be found using the formula: \ \ \[ S_n = a \frac{1-r^n}{1-r} \] \ where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms.
5Step 5: Substitute the Values
Plug \(a = -\frac{1}{2} \), \(r = -\frac{1}{2} \), and \(n = 5\) into the formula: \ \ \[ S_n = -\frac{1}{2} \frac{1-(-\frac{1}{2})^5}{1 + \frac{1}{2}} \]
6Step 6: Simplify the Sum
Calculate the numerator: \ \ \[ 1 - (-\frac{1}{2})^5 = 1 - (\frac{-1}{32}) = 1 + \frac{1}{32} = \frac{33}{32} \] \ and the denominator: \ \ \[ 1 + \frac{1}{2} = \frac{3}{2} = 1.5 \] so the equation becomes: \ \ \[ S_5 = -\frac{1}{2} \frac{\frac{33}{32}}{1.5} \]
7Step 7: Simplify Further
Performing the division in the formula: \ \ \[ -\frac{1}{2} \times \frac{33}{32 \times 1.5} = (-\frac{1}{2}) \times \frac{33}{48} = (-\frac{1}{2}) \times \frac{11}{16} = -\frac{11}{32} \]
Key Concepts
Finite Geometric SeriesCommon RatioSum FormulaNegative Exponents
Finite Geometric Series
A finite geometric series is a sequence of numbers with a fixed number of terms.
Each term in this series is obtained by multiplying the previous term by a constant value, known as the common ratio.
The series we are examining here is \(\sum_{i=1}^{5}(-2)^{-i}\), which counts as finite because it only goes from the first to the fifth term.
Understanding that it's 'finite' helps with identifying how we apply the sum formula.
Each term in this series is obtained by multiplying the previous term by a constant value, known as the common ratio.
The series we are examining here is \(\sum_{i=1}^{5}(-2)^{-i}\), which counts as finite because it only goes from the first to the fifth term.
Understanding that it's 'finite' helps with identifying how we apply the sum formula.
Common Ratio
The common ratio in a geometric series is the factor you multiply by to get from one term to the next.
In our series, each term is divided by \((-2)\), so the common ratio \((r)\) is \((-2)^{-1} \approx -\frac{1}{2}\).
For instance, if the first term is \(-\frac{1}{2}\), multiplying it by \(-\frac{1}{2}\) gives the second term, and so forth.
Identifying the common ratio is crucial for using the sum formula effectively.
In our series, each term is divided by \((-2)\), so the common ratio \((r)\) is \((-2)^{-1} \approx -\frac{1}{2}\).
For instance, if the first term is \(-\frac{1}{2}\), multiplying it by \(-\frac{1}{2}\) gives the second term, and so forth.
Identifying the common ratio is crucial for using the sum formula effectively.
Sum Formula
The sum of a finite geometric series can be calculated using a specific formula:
\[ S_n = a \frac{1-r^n}{1-r} \]
Here, \(a\) represents the first term of the series, \(r\) is the common ratio, and \(n\) is the number of terms.
In our exercise, \(a = -\frac{1}{2}\), \(r = -\frac{1}{2}\), and \(n = 5.\)
By substituting these values into the formula, you can find the sum of the series.
\[ S_n = a \frac{1-r^n}{1-r} \]
Here, \(a\) represents the first term of the series, \(r\) is the common ratio, and \(n\) is the number of terms.
In our exercise, \(a = -\frac{1}{2}\), \(r = -\frac{1}{2}\), and \(n = 5.\)
By substituting these values into the formula, you can find the sum of the series.
Negative Exponents
A negative exponent indicates the reciprocal of the base raised to the opposite positive exponent.
For example, \((-2)^{-1}\) is the same as \(\frac{1}{-2}\) or \(-\frac{1}{2}\).
Understanding this helps in identifying the correct values for the first term and the common ratio in the geometric series.
Calculations involving negative exponents are essential, especially when dealing with sequences like \(\sum_{i=1}^{5}(-2)^{-i}\).
For example, \((-2)^{-1}\) is the same as \(\frac{1}{-2}\) or \(-\frac{1}{2}\).
Understanding this helps in identifying the correct values for the first term and the common ratio in the geometric series.
Calculations involving negative exponents are essential, especially when dealing with sequences like \(\sum_{i=1}^{5}(-2)^{-i}\).
Other exercises in this chapter
Problem 11
Use the binomial theorem to expand each binomial. $$(x+1)^{3}$$
View solution Problem 11
List all terms of each finite sequence. \(c_{n}=(-2)^{n-1}\) for \(1 \leq n \leq 5\)
View solution Problem 12
Use the binomial theorem to expand each binomial. $$(y+1)^{4}$$
View solution Problem 12
List all terms of each finite sequence. \(c_{n}=(-3)^{n-2}\) for \(1 \leq n \leq 5\)
View solution