Problem 55 - Algebra and Trigonometry

Question

\(55-60\) . Write the sum without using sigma notation. $$ \sum_{k=1}^{5} \sqrt{k} $$

Step-by-Step Solution

Answer

The sum is approximately 8.382.

Step 1: Understand the Sigma Notation

Sigma notation is used to represent the summation of a sequence of numbers. In this exercise, it represents the sum of the square roots of numbers from 1 to 5.

Step 2: Identify the Sequence to Sum

The sequence given in the sigma notation is: \( \sqrt{1}, \sqrt{2}, \sqrt{3}, \sqrt{4}, \sqrt{5} \). We need to find the sum of these square roots.

Step 3: Calculate Each Square Root

Compute the square roots individually: \( \sqrt{1} = 1, \sqrt{2} \approx 1.414, \sqrt{3} \approx 1.732, \sqrt{4} = 2, \sqrt{5} \approx 2.236 \).

Step 4: Sum the Values

Add the calculated square root values together: \( 1 + 1.414 + 1.732 + 2 + 2.236 = 8.382 \)

Key Concepts

Square Roots

Square roots are mathematical functions that return the value which, when multiplied by itself, results in the original number. In simpler terms, the square root of a number \( n \) is a value \( x \) such that \( x^2 = n \). Calculating the square roots of numbers can be intuitive for perfect squares (e.g., the square root of 4 is 2), but it becomes slightly more complex for non-perfect squares. Here are some points to remember about square roots:

• The square root of a perfect square is always an integer.
• Non-perfect squares result in irrational numbers, often represented as decimals.
• Square roots are denoted with the radical sign \( \sqrt{} \), applied to the number you are evaluating.
• In expressions like \( \sqrt{4} \) or \( \sqrt{16} \), calculating the result involves recognizing simple multiplication patterns.

Understanding square roots is essential, especially in contexts like the given exercise where you sum them over a series.

Sequence Sum

A sequence sum involves adding up a finite list of numbers according to a specific rule or pattern. It's the core operation behind many mathematical analyses. In this exercise, the sequence was created using the sigma notation, which is a shorthand form to express the summation process for sequences. Here are some key points about sequence sums:

• Sigma notation \( \sum_{k=1}^{n} a_k \) describes the sum of a sequence \( a_1, a_2, ..., a_n \).
• The example from the exercise, \( \sum_{k=1}^{5} \sqrt{k} \), involves summing up the square roots of the first five natural numbers.
• To solve such an expression without sigma notation, you first identify and evaluate each term in the sequence individually.
• When all elements are calculated, sum them to find the total, like in the example: \( x_1 + x_2 + x_3 + x_4 + x_5 = 8.382 \).

Breaking down sigma notation into its sequence components can demystify the calculation, making it easier to follow.

Mathematical Computation

Mathematical computation refers to the process of performing operations to solve problems, calculate figures or derive results. Computations can include evaluating functions, solving equations, and summing sequences. This is often supported by manual calculation or computational tools. Let's explore some aspects:

• Understanding order of operations is vital, as it determines the sequence of computations.
• In our example, we computed the square roots before adding them, following this precedence.
• Throughout computations, approximations are sometimes necessary, especially with irrational numbers.
• Technology, like calculators or computational software, can greatly aid in these processes, ensuring accuracy and efficiency.

For students practicing mathematical computation, mastering these skills is crucial for accuracy and understanding in mathematical problem-solving. The exercise exemplifies this by clearly illustrating each step from root calculation to the sum total, reinforcing comprehension through tangible computation.