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Magazine Chapter 10: Problem 7
In Problems \(7-10,\) write out the terms of the given sum. $$ \sum_{k=1}^{5} \sqrt{k} $$
Short Answer
Expert verified
The terms are 1, \(\sqrt{2}\), \(\sqrt{3}\), 2, and \(\sqrt{5}\).
Step by step solution
01
Understand the Sum Notation
The sum notation \( \sum_{k=1}^{5} \sqrt{k} \) represents the sum of square roots of integers from 1 to 5. This means we will calculate \( \sqrt{1} \), \( \sqrt{2} \), \( \sqrt{3} \), \( \sqrt{4} \), and \( \sqrt{5} \), and then add them together.
02
Determine the First Term
The first term of the series is when \( k = 1 \). So, \( \sqrt{1} = 1 \).
03
Determine the Second Term
The second term of the series is when \( k = 2 \). So, \( \sqrt{2} \).
04
Determine the Third Term
The third term of the series is when \( k = 3 \). So, \( \sqrt{3} \).
05
Determine the Fourth Term
The fourth term of the series is when \( k = 4 \). So, \( \sqrt{4} = 2 \).
06
Determine the Fifth Term
The fifth and final term of the series is when \( k = 5 \). So, \( \sqrt{5} \).
07
Write Out All Terms
Listing all terms derived from Steps 2-6, the sum is \( 1 + \sqrt{2} + \sqrt{3} + 2 + \sqrt{5} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Square Roots
Square roots are special mathematical operations that reverse the process of squaring a number. When you take the square of a number, you're multiplying it by itself. For example, the square of 3 is 9 because 3 times 3 equals 9. The square root operation finds the original number from the squared result, and it's represented by the symbol \( \sqrt{} \). Hence, \( \sqrt{9} = 3 \).A few key points to remember about square roots:
- The square root of 1 is 1 because 1 times 1 equals 1.
- The square root of 4 is 2, which is a whole number, called a 'perfect square.'
- Most numbers, like 2 and 3, don't have neat square roots, so we leave them as square roots or use decimal approximations like \( 1.414 \) and \( 1.732 \) for \( \sqrt{2} \) and \( \sqrt{3} \) respectively.
Series
A series in mathematics is a way to describe the sum of terms from a sequence. When you use sum notation, written as \( \sum \), it's a way to signify adding up terms in a sequence.The expression \( \sum_{k=1}^{5} \sqrt{k} \) specifies a series where the terms are the square roots of consecutive integers starting from 1 up to 5. The series is understood by calculating each term:
- Begin with \( k = 1 \), which gives the first term \( \sqrt{1} = 1 \).
- Proceed to \( k = 2 \), the second term is \( \sqrt{2} \).
- For \( k = 3 \), the third term is \( \sqrt{3} \).
- Increase to \( k = 4 \), where the fourth term is \( \sqrt{4} = 2 \).
- Finally, at \( k = 5 \), conclude with \( \sqrt{5} \).
Sequences
Sequences are ordered lists of numbers that follow a specific pattern or rule. In mathematics, sequences help us understand how numbers are arranged and how they progress.For example, in the problem, the sequence begins with \( \sqrt{1} \), moves to \( \sqrt{2} \), followed by \( \sqrt{3} \), on to \( \sqrt{4} \), and finally \( \sqrt{5} \). This sequence is built upon consecutive integers beginning from 1 and following an arithmetic order.Key concepts related to sequences include:
- A sequence can be finite or infinite. Here, it's finite as it ends at 5.
- Sequences can include fractional, decimal, or even irrational terms derived from operations like square roots.
- Identifying the rule or pattern in a sequence, like square rooting each integer, is crucial to solving many math problems.
