Evaluate the Summation sum from i=1 to 14 of i^2
The given problem is asking for the calculation of a finite summation. Specifically, it's requesting the evaluation of the sum of the squares of the first fourteen positive integers. The notation indicates that i is the variable of summation, commencing at 1 and increasing by increments of 1 up to and including 14. The expression i^2 represents the square of each integer in this range. Thus, the problem requires determining the sum of 1^2 + 2^2 + 3^2 + ... + 13^2 + 14^2.
$\sum_{i = 1}^{14} i^{2}$
Answer
Solution:
Step 1:
Utilize the standard formula for the sum of squares of the first $n$ natural numbers:
$$\sum_{i = 1}^{n} i^{2} = \frac{n(n + 1)(2n + 1)}{6}$$
Step 2:
Insert the upper limit of the summation into the formula:
$$\frac{14(14 + 1)(2 \cdot 14 + 1)}{6}$$
Step 3:
Commence simplification.
Step 3.1:
Identify and eliminate common factors between numerator and denominator.
Step 3.1.1:
Extract the factor of $2$ from the numerator:
$$\frac{2 \cdot 7(14 + 1)(2 \cdot 14 + 1)}{6}$$
Step 3.1.2:
Remove shared factors.
Step 3.1.2.1:
Factor $2$ from the denominator:
$$\frac{2 \cdot 7(14 + 1)(2 \cdot 14 + 1)}{2 \cdot 3}$$
Step 3.1.2.2:
Cancel out the common factor of $2$:
$$\frac{\cancel{2} \cdot 7(14 + 1)(2 \cdot 14 + 1)}{\cancel{2} \cdot 3}$$
Step 3.1.2.3:
Reformulate the expression:
$$\frac{7(14 + 1)(2 \cdot 14 + 1)}{3}$$
Step 3.2:
Further simplify the numerator.
Step 3.2.1:
Calculate $2 \cdot 14$:
$$\frac{7(14 + 1)(28 + 1)}{3}$$
Step 3.2.2:
Sum up $14 + 1$:
$$\frac{7 \cdot 15(28 + 1)}{3}$$
Step 3.2.3:
Multiply $7$ by $15$:
$$\frac{105(28 + 1)}{3}$$
Step 3.2.4:
Combine $28 + 1$:
$$\frac{105 \cdot 29}{3}$$
Step 3.3:
Finalize the simplification.
Step 3.3.1:
Multiply $105$ by $29$:
$$\frac{3045}{3}$$
Step 3.3.2:
Divide $3045$ by $3$ to get the final result:
$$1015$$
Knowledge Notes:
The problem involves evaluating the sum of squares of the first $n$ natural numbers. The relevant knowledge points include:
Summation Notation: The sigma notation $\sum$ is used to denote the sum of a sequence of numbers. In this problem, we are summing the squares of the first $n$ natural numbers, which is represented as $\sum_{i = 1}^{n} i^{2}$.
Sum of Squares Formula: The formula for the sum of the squares of the first $n$ natural numbers is given by:
$$\sum_{i = 1}^{n} i^{2} = \frac{n(n + 1)(2n + 1)}{6}$$ This is a well-known result in mathematics and is used to quickly calculate the sum without having to add each square individually.
Algebraic Simplification: The process involves algebraic manipulation to simplify the expression. This includes factoring out common factors, canceling terms, and performing arithmetic operations such as multiplication and addition.
Arithmetic Operations: Basic arithmetic operations鈥攁ddition, multiplication, and division鈥攁re used to simplify the expression to get the final result.
Understanding these concepts is crucial for solving summation problems efficiently and is a fundamental part of algebra and discrete mathematics.
