91Ó°ÊÓ

Summation is a fundamental concept in mathematics that refers to the addition of a sequence of numbers. The summation notation is represented by the Greek letter sigma \(( \sum )\). In the given exercise, we are asked to sum multiples of 4, specifically for values of n from 1 to 3. We write this as \(\sum_{i=1}^{n} 4i\), which means we add up 4 times each integer from 1 to n.
For instance, when n=3, we compute \(\sum_{i=1}^{3} 4i = 4 \times 1 + 4 \times 2 + 4 \times 3\), which yields 4 + 8 + 12 = 24. This summation process lets us systematically add up terms in the series.

A mathematical series is the sum of the terms of a sequence. In our case, the series we are working with is composed of terms that are multiples of 4. When dealing with series, it's important to clearly distinguish between a series and a sequence. A sequence is an ordered list of numbers, while a series is the summation of a sequence's terms.
Here, the sequence is \( 4, 8, 12, ... \), and its corresponding series for n=1, 2, and 3 are:

• n=1: 4
• n=2: 4 + 8 = 12
• n=3: 4 + 8 + 12 = 24

This analysis helps understand whether a given expression, such as \(( 8n - 4)\) in the exercise, accurately represents the sum of the series. For n=3, the series sum 24 does not match 8n-4, indicating a discrepancy and leading to a deeper examination of the series behavior.

Algebraic equations are mathematical statements that show the equality between two expressions. In the context of the given problem, we are comparing the summation series with an algebraic expression: \(( 8n - 4)\). To check the validity for particular values of n (1, 2, and 3), we evaluate both the summation series and the algebraic expression independently and then compare them.
For n=1:
Summation: \(\sum_{i=1}^{1} 4i = 4\)
Algebraic expression: \(8 \times 1 - 4 = 4\)
Both sides are equal, so the equation holds for n=1.
For n=2:
Summation: \( 4 + 8 = 12\)
Algebraic expression: \(8 \times 2 - 4 = 12\)
Again, both sides are equal.
For n=3:
Summation: \( 4 + 8 + 12 = 24\)
Algebraic expression: \(8 \times 3 - 4 = 20\)
In this case, both sides are not equal, indicating the algebraic equation does not hold.
This step-by-step evaluation demonstrates how algebra can be applied to verify mathematical series and identify inconsistencies.