Evaluate the Summation sum from j=1 to 5 of 4j-16
This question is asking for the evaluation of a finite summation. Specifically, you need to find the total when you add up the values generated by the expression "4j - 16" as the variable "j" takes on each integer value from 1 to 5. The process involves calculating the expression for each value of "j," and then summing all those results to find the total sum.
$\sum_{j = 1}^{5} 4 j - 16$
Answer
Solution:
Step 1: Expansion of the Summation
Write out the terms of the summation for each integer value of $j$ from 1 to 5.
$$(4 \cdot 1 - 16) + (4 \cdot 2 - 16) + (4 \cdot 3 - 16) + (4 \cdot 4 - 16) + (4 \cdot 5 - 16)$$
Step 2: Calculate the Sum
Perform the addition of the terms to find the total sum.
$$= -12 + (-8) + (-4) + 0 + 4$$ $$= -20$$
Knowledge Notes:
The problem involves evaluating a finite summation, which is a process of adding up a sequence of numbers generated by a formula involving an index variable. In this case, the index variable is $j$, and it ranges from 1 to 5. The formula given is $4j - 16$, which means we multiply $j$ by 4 and subtract 16 for each value of $j$ within the range.
To solve this problem, we follow these steps:
Expansion of the Summation: We start by expanding the summation into its individual terms. This means we substitute each integer value of $j$ into the formula and write out the result.
Calculate the Sum: Once we have the expanded form, we add up all the terms to find the total sum. This step may involve simplifying each term before performing the addition.
In this problem, the summation simplifies to a single number, which is the final result of the summation.
Relevant knowledge points include:
Summation Notation: Also known as sigma notation, it is a way to write a sum of many similar terms in a concise form. The Greek letter sigma ($\Sigma$) is used to denote the sum.
Arithmetic Operations: The problem requires basic arithmetic operations like multiplication and addition.
Algebraic Simplification: This involves combining like terms and simplifying expressions according to algebraic rules.
Finite Series: A finite series is a sum of a finite number of terms. In this case, the series has 5 terms corresponding to the values of $j$ from 1 to 5.
Understanding these concepts is essential for solving summation problems in mathematics.
