Evaluate the Summation sum from J=1 to 6 of 9J^2

Solution:

Step 1

Extract the constant $9$ from the summation: $9 \sum_{J = 1}^{6} J^{2}$.

Step 2

Apply the sum of squares formula: $\sum_{k = 1}^{n} k^{2} = \frac{n(n + 1)(2n + 1)}{6}$.

Step 3

Insert the upper limit of the summation into the formula, and don't forget to include the constant we factored out: $9 \left( \frac{6(6 + 1)(2 \cdot 6 + 1)}{6} \right)$.

Step 4

Proceed to simplify the expression.

Step 4.1

First, simplify the numerator.

Step 4.1.1

Add together $6$ and $1$: $9 \cdot \frac{6 \cdot 7 (2 \cdot 6 + 1)}{6}$.

Step 4.1.2

Perform the exponentiation operations.

Step 4.1.2.1

Multiply $6$ by $7$: $9 \cdot \frac{42 (2 \cdot 6 + 1)}{6}$.

Step 4.1.2.2

Multiply $2$ by $6$: $9 \cdot \frac{42 (12 + 1)}{6}$.

Step 4.1.3

Combine $12$ and $1$: $9 \cdot \frac{42 \cdot 13}{6}$.

Step 4.2

Next, simplify the terms.

Step 4.2.1

Multiply $42$ by $13$: $9 \left( \frac{546}{6} \right)$.

Step 4.2.2

Eliminate the common factor of $3$.

Step 4.2.2.1

Extract $3$ from $9$: $3 \cdot 3 \cdot \frac{546}{6}$.

Step 4.2.2.2

Extract $3$ from $6$: $3 \cdot 3 \cdot \frac{546}{3 \cdot 2}$.

Step 4.2.2.3

Cancel out the common factor of $3$: $3 \cdot \frac{546}{2}$.

Step 4.2.2.4

Rewrite the simplified expression: $3 \left( \frac{546}{2} \right)$.

Step 4.2.3

Combine $3$ with $\frac{546}{2}$: $\frac{3 \cdot 546}{2}$.

Step 4.2.4

Final simplification of the expression.

Step 4.2.4.1

Multiply $3$ by $546$: $\frac{1638}{2}$.

Step 4.2.4.2

Divide $1638$ by $2$: $819$.

Knowledge Notes:

The problem involves evaluating a summation of the form $\sum_{J=1}^{6} 9J^2$. To solve this, we use the following knowledge points:

1. Factorization: We can factor out constants from a summation to simplify the expression. In this case, we factored out the constant $9$.

2. Sum of Squares Formula: The sum of the squares of the first $n$ natural numbers is given by the formula $\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$. This formula is derived from mathematical induction or other advanced techniques in combinatorics.

3. Substitution: After factoring out constants and identifying the correct formula, we substitute the upper limit of the summation into the formula.

4. Simplification: The process involves simplifying the algebraic expression step by step, including operations like addition, multiplication, and canceling out common factors.

5. Arithmetic Operations: Basic arithmetic operations such as addition, multiplication, and division are used to simplify the expression to its final form.

By following these steps and applying the relevant knowledge, we can efficiently evaluate the given summation.