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Magazine Chapter 5: Problem 37
Calculate the indicated partial sums. $$\sum_{i=1}^{4}(1.7+0.04 i)$$
Short Answer
Expert verified
The partial sum is 7.20.
Step by step solution
01
Understand the Series
We are tasked with calculating the sum of the series \( \sum_{i=1}^{4}(1.7+0.04 i) \). Here, \( i \) is the index of summation, and it runs from 1 to 4. The expression inside the sum, \( 1.7+0.04i \), represents each term in the series that we need to add.
02
Calculate Each Term
Substitute \( i = 1, 2, 3, \) and \( 4 \) into the expression \( 1.7 + 0.04i \) to find each term of the series.- For \( i = 1 \): \( 1.7 + 0.04 \times 1 = 1.7 + 0.04 = 1.74 \).- For \( i = 2 \): \( 1.7 + 0.04 \times 2 = 1.7 + 0.08 = 1.78 \).- For \( i = 3 \): \( 1.7 + 0.04 \times 3 = 1.7 + 0.12 = 1.82 \).- For \( i = 4 \): \( 1.7 + 0.04 \times 4 = 1.7 + 0.16 = 1.86 \).
03
Sum the Terms
Add the terms we calculated:\[ 1.74 + 1.78 + 1.82 + 1.86 \].Sum these step-by-step:- \(1.74 + 1.78 = 3.52 \).- \(3.52 + 1.82 = 5.34 \).- \(5.34 + 1.86 = 7.20 \).
04
Conclusion
The sum of the series \( \sum_{i=1}^{4}(1.7+0.04 i) \) is 7.20.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Series Calculation
A series is essentially the sum of the terms of a sequence. When calculating a series, the main goal is to determine the total after all the terms in the sequence have been added together. In the case of our example, we have a simple arithmetic sequence and we aim to find its partial sum.
**Partial Sums**
**Partial Sums**
- A partial sum refers to adding only a subset of terms from the entire sequence. For example, adding from the first term to the fourth term, as in the provided exercise, is a partial sum.
- This step-by-step addition helps in understanding how the total sum is constructed out of individual terms of the sequence.
- By calculating each term and adding them consecutively, we form our series' sum: for example, 1.74 + 1.78 + 1.82 + 1.86 equals 7.20.
Sequence and Series
To understand series, it's vital to recognize what sequence they stem from. A sequence is an ordered list of numbers which follow a particular pattern or rule. Each individual element of the sequence is referred to as a term.
**Components of a Sequence**
**Components of a Sequence**
- The sequence given in our problem is defined by the formula: \( 1.7 + 0.04i \). This is a simple algebraic expression where each term is calculated by substituting consecutive integer values of \( i \).
- For example, substituting \( i = 1 \), we get the first term as 1.74. Continuing this substitution gives us all terms up to the fourth one, 1.86 in this case.
- Once we have our sequence, transforming it into a series involves summing these sequential terms. This conversion is what marks the transition from a sequence to a series.
- Remember, in mathematics, especially in calculus, sequences and series lay foundational concepts that lead into functions and analysis.
Mathematical Summation
Summation is a mathematical operation that involves adding a sequence of numbers. This operation is often encountered in series calculations and is represented by the Greek letter \( \Sigma \) (Sigma).
**Understanding Sigma Notation**
**Understanding Sigma Notation**
- The Sigma notation, \( \sum_{i=a}^{b} f(i) \), is a concise way to express the sum of terms defined by the function \( f(i) \) as \( i \) ranges from \( a \) to \( b \).
- It symbolically indicates everything expanded out: \( f(a) + f(a+1) + ... + f(b) \).
- In the exercise, the summation expression \( \sum_{i=1}^{4}(1.7+0.04i) \) shows that our task is to evaluate the sum starting from \( i = 1 \) up to \( i = 4 \).
- This method of representing summation simplifies the expression of long and potentially complex summations into manageable terms when performing calculations.
