Evaluate the Summation sum from i=4 to 8 of 3i-2

Solution:

Step 1:

Write out the terms of the summation for each integer value of $i$ from 4 to 8.

$3 \cdot 4 - 2 + 3 \cdot 5 - 2 + 3 \cdot 6 - 2 + 3 \cdot 7 - 2 + 3 \cdot 8 - 2$

Step 2:

Calculate the sum of the terms.

$80$

Knowledge Notes:

The problem asks for the evaluation of a finite summation. A summation is a mathematical notation used to represent the addition of a sequence of numbers. The specific summation in question is defined as the sum of terms of the form $3i - 2$ for each integer $i$ ranging from 4 to 8.

To solve this, we can follow these steps:

1. Expansion of the Series: This involves writing out each term of the sequence explicitly. For a summation, this means substituting each integer value of $i$ within the given range into the expression $3i - 2$.

2. Simplification: After expanding the series, we combine like terms or perform any necessary arithmetic operations to simplify the expression to a single numerical value.

In this case, the summation can be expanded as follows:

$$ \sum_{i=4}^{8} (3i - 2) = (3 \cdot 4 - 2) + (3 \cdot 5 - 2) + (3 \cdot 6 - 2) + (3 \cdot 7 - 2) + (3 \cdot 8 - 2) $$

Each term is then simplified by performing the multiplication and subtraction:

$$ (12 - 2) + (15 - 2) + (18 - 2) + (21 - 2) + (24 - 2) $$

Finally, we add up the simplified terms to get the sum:

$$ 10 + 13 + 16 + 19 + 22 = 80 $$

The result of the summation is 80.

Relevant knowledge points include understanding the notation and concept of a summation, arithmetic operations (addition, subtraction, multiplication), and the ability to simplify expressions.