Evaluate the Summation sum from j=1 to 6 of 5j^2
The problem asks to calculate the total sum of a series where the term being summed is a function of the variable 'j'. This function is given as 5j^2, which implies that each term is five times the square of the index 'j'. The summation process involves computing this term for each integer value of 'j' starting at 1 and ending at 6, and then adding up all of those computed values to get a final result. In other words, you are required to find the combined value of 5(1^2) + 5(2^2) + 5(3^2) + 5(4^2) + 5(5^2) + 5(6^2).
$\sum_{j = 1}^{6} 5 j^{2}$
Answer
Solution:
Step 1:
Extract the constant $5$ from the summation: $5 \sum_{j = 1}^{6} j^{2}$.
Step 2:
Use 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 multiply by the constant: $5 \left( \frac{6(6 + 1)(2 \cdot 6 + 1)}{6} \right)$.
Step 4:
Perform the simplification process.
Step 4.1:
Begin by simplifying the numerator.
Step 4.1.1:
Add together $6$ and $1$: $5 \left( \frac{6 \cdot 7(2 \cdot 6 + 1)}{6} \right)$.
Step 4.1.2:
Resolve the multiplication.
Step 4.1.2.1:
Calculate $6 \cdot 7$: $5 \left( \frac{42(2 \cdot 6 + 1)}{6} \right)$.
Step 4.1.2.2:
Calculate $2 \cdot 6$: $5 \left( \frac{42(12 + 1)}{6} \right)$.
Step 4.1.3:
Add $12$ and $1$: $5 \left( \frac{42 \cdot 13}{6} \right)$.
Step 4.2:
Finalize the simplification.
Step 4.2.1:
Multiply $42$ by $13$: $5 \left( \frac{546}{6} \right)$.
Step 4.2.2:
Divide $546$ by $6$: $5 \cdot 91$.
Step 4.2.3:
Multiply $5$ by $91$: $455$.
Knowledge Notes:
The problem involves evaluating a summation of a sequence of squared terms. Here are the relevant knowledge points:
Summation Notation: The summation notation $\sum$ is used to denote the sum of a sequence of terms. The expression $\sum_{j=1}^{6} 5j^2$ means to sum the terms $5j^2$ where $j$ ranges from $1$ to $6$.
Factoring Constants: In summation, constants can be factored out to simplify the expression. For example, $5 \sum_{j=1}^{6} j^2$ means $5$ times the sum of $j^2$ from $1$ to $6$.
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 methods of proof.
Arithmetic Operations: Simplifying expressions involves performing arithmetic operations such as addition, multiplication, and division. These operations follow the order of operations, which dictates the sequence in which the operations should be performed to correctly simplify an expression.
Algebraic Simplification: The process of simplifying algebraic expressions involves reducing them to their simplest form by performing arithmetic operations and combining like terms.
By applying these concepts, we can evaluate the given summation systematically and arrive at the final result.
