Evaluate the Summation sum from i=1 to 5 of 2 square root of i
The question is asking for the calculation of a specific summation. You're instructed to compute the total of a series where you take the square root of each integer i from 1 to 5, multiply that square root by 2, and then add up all those products to find the final sum. The notation you provided describes the mathematical process of summing (adding together) the values generated by the function 2√i for each integer value of i in the range from 1 to 5, inclusive.
$\sum_{i = 1}^{5} 2 \sqrt{i}$
Answer
Solution:
Step 1: Expansion of the Summation
Write out the terms of the summation explicitly for each value of $i$ from 1 to 5.
$2 \sqrt{1} + 2 \sqrt{2} + 2 \sqrt{3} + 2 \sqrt{4} + 2 \sqrt{5}$
Step 2: Calculation of the Sum
Compute the sum of the terms.
$2(1) + 2(\sqrt{2}) + 2(\sqrt{3}) + 2(2) + 2(\sqrt{5}) = 2 + 2\sqrt{2} + 2\sqrt{3} + 4 + 2\sqrt{5} \approx 16.76466469$
Knowledge Notes:
The problem involves evaluating a finite summation, which is a process of adding up a sequence of numbers resulting from applying a function to a range of integers. Here, the function is $f(i) = 2\sqrt{i}$, and the range is from 1 to 5. The steps to solve this problem are:
Expansion of the Summation: This involves writing out each term of the summation explicitly. In this case, the summation is expanded by substituting the values of $i$ from 1 to 5 into the function $f(i)$.
Calculation of the Sum: After expanding the summation, the next step is to calculate the sum of all terms. This involves performing the arithmetic operations indicated by the function. In this problem, it means calculating the square root of each integer, multiplying by 2, and then adding all the results together.
Relevant knowledge points include:
Summation Notation: The summation notation $\sum$ is used to represent the sum of a sequence of terms. The expression under the summation sign indicates the general term of the sequence, while the limits of summation (in this case, $i=1$ to $i=5$) indicate the range over which the summation is to be carried out.
Square Root: The square root of a number $x$, denoted as $\sqrt{x}$, is a value that, when multiplied by itself, gives the number $x$. Square roots can be both positive and negative, but in this context, we are dealing with the principal (non-negative) square root.
Arithmetic Operations: The basic arithmetic operations involved in evaluating the summation are multiplication and addition. The multiplication of the square root by 2 is done first, followed by the addition of the terms.
Approximation: When dealing with irrational numbers (like $\sqrt{2}$ and $\sqrt{3}$), it is often necessary to approximate the value to a certain number of decimal places, as these numbers cannot be expressed exactly in decimal form.
Calculator Use: For the actual computation of the square roots and the subsequent arithmetic, a calculator or computational software is typically used to obtain an approximate decimal value.
