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Magazine Chapter 5: Problem 2
Compute \(\sum_{j=0}^{3}\left(j^{2}-j\right)\) and \(\sum_{j=1}^{n} 1 / 2^{j}\)
Short Answer
Expert verified
The first sum is 8, and the second sum is \(1 - \frac{1}{2^n}\).
Step by step solution
01
Identify the First Series
We need to compute the sum \(\sum_{j=0}^{3}(j^2 - j)\). This means we will calculate each term from \(j=0\) to \(j=3\), then add them together.
02
Calculate Each Term of the First Series
Calculate each term individually: \(j=0: 0^2 - 0 = 0\), \(j=1: 1^2 - 1 = 0\), \(j=2: 2^2 - 2 = 2\), \(j=3: 3^2 - 3 = 6\).
03
Sum the Terms of the First Series
Now add the terms: \(0 + 0 + 2 + 6 = 8\). So, \(\sum_{j=0}^{3}(j^2 - j) = 8\).
04
Identify and Analyze the Second Series
We need to compute the sum \(\sum_{j=1}^{n} \frac{1}{2^j}\). It's a geometric series with the first term \(a = \frac{1}{2}\) and common ratio \(r = \frac{1}{2}\).
05
Use the Formula for Geometric Series
The sum of the first \(n\) terms of a geometric series is \( S = \frac{a(1 - r^n)}{1 - r} \). Substitute \(a = \frac{1}{2}\) and \(r = \frac{1}{2}\): \[ S = \frac{\frac{1}{2}(1 - (\frac{1}{2})^n)}{1 - \frac{1}{2}} = 1 - \frac{1}{2^n} \]
06
Conclusion for the Second Series
Therefore, \(\sum_{j=1}^{n} \frac{1}{2^j} = 1 - \frac{1}{2^n}\). This gives the sum of the series for any \(n\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Series Summation
Series summation is the process of adding up all the terms in a series. A series is a sequence of numbers where each term is connected by a specific rule. In the exercise, we tackled two series. The first series involved summing terms calculated using the formula \(j^2 - j\), from \(j = 0\) to \(j = 3\). Here鈥檚 how it works:
Understanding this process helps in grasping complex mathematical concepts systematically.
- Find each term using the given formula.
- For example, when \(j = 0\), the term is \(0^2 - 0 = 0\); when \(j = 1\), it鈥檚 \(1^2 - 1 = 0\).
- Add all these calculated terms together (like \(0 + 0 + 2 + 6\)) to reach the total sum, which in this case is 8.
Understanding this process helps in grasping complex mathematical concepts systematically.
Geometric Progression
A geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. In the exercise, the second series is a geometric progression where each term is of the form \(\frac{1}{2^j}\). Let's break it down:
They are essential in understanding patterns in growth and decay models, like population growth or radioactive decay.
- The first term \(a\) is \(\frac{1}{2}\).
- The common ratio \(r\) is also \(\frac{1}{2}\).
- Each subsequent term is simply half of the previous term.
They are essential in understanding patterns in growth and decay models, like population growth or radioactive decay.
Mathematical Series
A mathematical series is the sum of the terms of a sequence. It can be finite or infinite. The two series from the exercise are examples of finite series as they have a definite number of terms to add. Here鈥檚 a general approach to tackling mathematical series:
Studying them enhances solving skills for real-world problems related to summation.
- Identify the pattern or rule that defines the sequence.
- Calculate each individual term if necessary.
- Sum up all the terms to find the total, or use a formula for ease if available.
Studying them enhances solving skills for real-world problems related to summation.
Sums of Sequences
Sums of sequences involve the addition of all the numbers in a series derived from a sequence. In our exercise, after applying the rules for calculating each term, the next step was to add them up. Each sequence can be infinite with a rule that continues endlessly, or finite with a clear endpoint. Some important points:
This knowledge is directly applicable, not just in mathematics, but in fields like computer science, finance, and physics.
- Recognize whether the sequence is arithmetic or geometric, as this changes the approach.
- For geometric sequences, use the formula for the sum, which is often necessary when dealing with infinite sequences to manage a finite sum.
- In the case of \( \sum_{j=1}^{n} \frac{1}{2^j} \), understanding the geometric properties allows quick summation.
This knowledge is directly applicable, not just in mathematics, but in fields like computer science, finance, and physics.
