Evaluate the Summation sum from j=1 to 10 of 3j+5
The question asks for an evaluation of a finite arithmetic summation. Specifically, it requests that you find the total sum of the values generated by the given arithmetic expression, "3j+5," where "j" is the variable that sequentially takes on integer values from 1 to 10. For each value of "j," you would calculate the expression "3j+5" and then sum all of these computed values together to obtain the final result.
$\sum_{j = 1}^{10} 3 j + 5$
Answer
Solution:
Step 1
Decompose the given summation into two separate summations by distributing the summation operator over the addition inside the sum.
$\sum_{j = 1}^{10} (3j + 5) = 3\sum_{j = 1}^{10} j + \sum_{j = 1}^{10} 5$
Step 2
Calculate the first summation $3 \sum_{j = 1}^{10} j$.
Step 2.1
Utilize the arithmetic series sum formula:
$\sum_{k = 1}^{n} k = \frac{n(n + 1)}{2}$
Step 2.2
Insert the upper limit of the summation into the formula and multiply by the coefficient:
$3 \left( \frac{10(10 + 1)}{2} \right)$
Step 2.3
Perform the arithmetic operations:
Step 2.3.1
Combine $10$ and $1$:
$3 \frac{10 \cdot 11}{2}$
Step 2.3.2
Multiply $10$ by $11$:
$3 \left( \frac{110}{2} \right)$
Step 2.3.3
Divide $110$ by $2$:
$3 \cdot 55$
Step 2.3.4
Multiply $3$ by $55$:
$165$
Step 3
Compute the second summation $\sum_{j = 1}^{10} 5$.
Step 3.1
Apply the formula for the summation of a constant term:
$\sum_{k = 1}^{n} c = cn$
Step 3.2
Plug in the values into the formula:
$5(10)$
Step 3.3
Carry out the multiplication of $5$ and $10$:
$50$
Step 4
Combine the results of the two summations:
$165 + 50$
Step 5
Sum the two numbers to find the final result:
$215$
Knowledge Notes:
To solve the given problem, we use the properties of summation and formulas for specific types of series. The relevant knowledge points include:
Summation Properties: Summation is a linear operator, which means that the summation of a sum can be split into the summation of its addends. For example, $\sum (a_j + b_j) = \sum a_j + \sum b_j$.
Arithmetic Series: An arithmetic series is the sum of the terms of an arithmetic sequence. The sum of the first $n$ natural numbers is given by the formula $\sum_{k = 1}^{n} k = \frac{n(n + 1)}{2}$.
Summation of a Constant: The sum of a constant $c$ over $n$ terms is simply $n$ times the constant: $\sum_{k = 1}^{n} c = cn$.
Basic Arithmetic Operations: The steps involve basic arithmetic operations such as addition, multiplication, and division.
By applying these principles, we can break down the original summation into more manageable parts, calculate each part separately using the appropriate formulas, and then combine the results to get the final sum.
