Evaluate the Summation sum from i=1 to 5 of 6 square root of i
The problem asks for the calculation of a finite mathematical summation. Specifically, it requires evaluating the sum of a series of terms where each term is the product of 6 and the square root of the index i. The index i ranges from 1 to 5, meaning you will be summing five terms. Each of these terms will be of the form 6√i, where i takes on the values 1 through 5 inclusively in consecutive order.
$\sum_{i = 1}^{5} 6 \sqrt{i}$
Answer
Solution:
Step 1: Expansion of the Summation
Write out the terms of the summation for each integer value of $i$ from 1 to 5.
$6 \sqrt{1} + 6 \sqrt{2} + 6 \sqrt{3} + 6 \sqrt{4} + 6 \sqrt{5}$
Step 2: Calculation of the Sum
Compute the sum of the terms.
$6(1) + 6(\sqrt{2}) + 6(\sqrt{3}) + 6(2) + 6(\sqrt{5}) = 6 + 6\sqrt{2} + 6\sqrt{3} + 12 + 6\sqrt{5} \approx 50.29399408$
Knowledge Notes:
To solve the given problem, we need to understand the concept of summation and the properties of square roots.
Summation (危): This is a mathematical notation used to represent the addition of a sequence of numbers. The summation symbol (危) is followed by an expression that represents the terms in the series. The variable below the symbol indicates the starting index, and the number above indicates the ending index.
Square Root: The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because 3 times 3 equals 9.
Simplification: In the context of this problem, simplification involves computing the square roots of the numbers and multiplying them by 6, as indicated by the summation expression.
Approximation: Since some square roots do not result in whole numbers, we may need to approximate the value of the summation to a certain number of decimal places.
Arithmetic Operations: Basic arithmetic operations are used to add the terms together once they have been simplified.
In this problem, we expanded the summation into individual terms, computed the square roots where necessary, multiplied each by 6, and then added the results to find the approximate sum.
