Evaluate the Summation sum from i=1 to 5 of 5 square root of i

Solution:

Step 1: Write out the terms of the summation

List each term of the summation by substituting the values of $i$ from 1 to 5 into the expression $5\sqrt{i}$.

$5\sqrt{1} + 5\sqrt{2} + 5\sqrt{3} + 5\sqrt{4} + 5\sqrt{5}$

Step 2: Calculate the sum of the terms

Add up the terms to find the total sum.

The sum is approximately $41.91166173$.

Knowledge Notes:

To solve the given problem, we need to understand the concept of summation notation and the properties of square roots.

• Summation Notation: The summation notation $\sum$ is used to represent the sum of a sequence of terms. The expression $\sum_{i=1}^{n} a_i$ means that we should sum all terms $a_i$ from $i=1$ to $i=n$.

• Square Roots: 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 of perfect squares are integers, while square roots of non-perfect squares are irrational numbers.

• Arithmetic Operations: When adding terms that involve square roots, we can only combine like terms (terms with the same radicand). In this problem, since each term has a different radicand, we cannot combine them algebraically. Instead, we calculate the numerical value of each term and then add them together.

• Approximation: When dealing with irrational numbers, we often use approximations to express our answer in decimal form. This is because irrational numbers cannot be expressed as a finite or repeating decimal.

In this problem, we first expand the summation by writing out each term for $i$ ranging from 1 to 5. Then, we calculate the square root of each value of $i$, multiply it by 5, and add up all the terms to get the final sum. The result is an approximation because some of the square roots are of non-perfect squares, leading to irrational numbers.