Evaluate the Summation sum from k=1 to 21 of k^2
The question asks for a mathematical evaluation of a series where each term is the square of an integer k, and specifically where k ranges from 1 to 21. It involves performing an operation known as summation, where you successively add together the squares of these integers, starting with 1 squared (1^2), then adding 2 squared (2^2), then 3 squared (3^2), and so on, all the way up to 21 squared (21^2). The problem is looking for a single numerical value that represents the total of all these squared terms added together.
$\sum_{k = 1}^{21} k^{2}$
Answer
Solution:
Step 1:
Utilize the standard formula for the sum of squares of the first $n$ natural numbers: $\sum_{k = 1}^{n} k^{2} = \frac{n(n + 1)(2n + 1)}{6}$.
Step 2:
Insert the upper limit of the summation into the formula: $\frac{21(21 + 1)(2 \cdot 21 + 1)}{6}$.
Step 3:
Commence simplification.
Step 3.1:
Identify and eliminate common factors between numerator and denominator.
Step 3.1.1:
Extract the common factor of $3$ from the numerator: $\frac{3(7(21 + 1)(2 \cdot 21 + 1))}{6}$.
Step 3.1.2:
Proceed to cancel out the common factors.
Step 3.1.2.1:
Factor out $3$ from the denominator: $\frac{3(7(21 + 1)(2 \cdot 21 + 1))}{3 \cdot 2}$.
Step 3.1.2.2:
Eliminate the common factor of $3$: $\frac{\cancel{3}(7(21 + 1)(2 \cdot 21 + 1))}{\cancel{3} \cdot 2}$.
Step 3.1.2.3:
Reformulate the expression: $\frac{7(21 + 1)(2 \cdot 21 + 1)}{2}$.
Step 3.2:
Further simplify the numerator.
Step 3.2.1:
Double the value of $21$: $\frac{7(21 + 1)(42 + 1)}{2}$.
Step 3.2.2:
Combine $21$ and $1$: $\frac{7 \cdot 22(42 + 1)}{2}$.
Step 3.2.3:
Multiply $7$ by $22$: $\frac{154(42 + 1)}{2}$.
Step 3.2.4:
Sum up $42$ and $1$: $\frac{154 \cdot 43}{2}$.
Step 3.3:
Finalize the simplification.
Step 3.3.1:
Calculate the product of $154$ and $43$: $\frac{6622}{2}$.
Step 3.3.2:
Divide $6622$ by $2$: $3311$.
Knowledge Notes:
The problem involves evaluating the sum of the squares of the first $21$ natural numbers. This is a classic summation problem that can be solved using a well-known formula for the sum of squares of the first $n$ natural numbers.
Relevant knowledge points include:
Summation Notation: Summation notation (denoted by the Greek letter sigma, $\sum$) is a way to represent the sum of a sequence of terms. It is often used in mathematics to condense the notation of adding up a long sequence of numbers.
Sum of Squares Formula: The formula for the sum of the squares of the first $n$ natural numbers is $\sum_{k = 1}^{n} k^{2} = \frac{n(n + 1)(2n + 1)}{6}$. This is a closed-form expression that allows for the quick calculation of the sum without having to add each term individually.
Simplification Techniques: Simplifying expressions often involves factoring out common factors, canceling terms, and performing arithmetic operations such as addition, multiplication, and division.
Arithmetic Operations: Basic arithmetic operations include addition, subtraction, multiplication, and division. These operations are fundamental to simplifying mathematical expressions.
Understanding these concepts is crucial for solving summation problems efficiently and accurately.
