Evaluate the Summation sum from j=1 to 5 of 3j-7
The question is asking to perform a summation calculation, which is a mathematical operation where you add up the values of a sequence. Specifically, the question requires you to start at the first term (where j=1) and continue adding successive terms up until the fifth term (where j=5). The sequence you are summing is given by the formula 3j-7, which means for each term of the sequence, you multiply the current value of j by 3 and then subtract 7. You need to calculate this expression for each value of j from 1 to 5 and then add all these calculated values together to get the final sum.
$\sum_{j = 1}^{5} 3 j - 7$
Answer
Solution:
Step 1:
Write out the terms of the summation for each integer value of $j$ from 1 to 5.
$3 \cdot 1 - 7 + 3 \cdot 2 - 7 + 3 \cdot 3 - 7 + 3 \cdot 4 - 7 + 3 \cdot 5 - 7$
Step 2:
Calculate the sum of the terms.
$-4 + (-1) + 2 + 5 + 8 = 10$
Knowledge Notes:
The problem involves evaluating a finite summation, which is a common operation in mathematics where you sum the values of a sequence of terms. The given summation is $\sum_{j=1}^{5} (3j - 7)$, which means you need to calculate the sum of the terms $3j - 7$ for each integer value of $j$ from 1 to 5.
To solve this, you can follow these steps:
Expansion of the Series: This involves writing out each term of the sequence by substituting the values of $j$ into the expression $3j - 7$. This step helps to visualize all the terms that will be added together.
Simplification: After expanding the series, you add up all the terms to find the total sum. This step may involve combining like terms and performing arithmetic operations.
In this case, the summation can be expanded as follows:
$3 \cdot 1 - 7 + 3 \cdot 2 - 7 + 3 \cdot 3 - 7 + 3 \cdot 4 - 7 + 3 \cdot 5 - 7$ This results in a series of arithmetic operations that can be simplified to find the final sum.
The relevant knowledge points for solving this problem include understanding of summation notation, arithmetic operations, and the ability to simplify expressions. Summation notation is a concise way to represent the sum of a sequence of terms, and it is widely used in various fields of mathematics, including algebra, calculus, and statistics. Arithmetic operations include addition, subtraction, multiplication, and division, which are fundamental in simplifying mathematical expressions.
