Problem 49 - Algebra 2

Question

Which expression represents the sum \(10+20+30+40 ?\) I. \(\sum_{n=1}^{4} 10 n\) II. \(\sum_{n=10}^{40} 10 n\) III. \(10\left(\sum_{n=1}^{4} n\right)\) A. I and II only B. I and III only C. II and III only D. \(1, \|\), and \(\| 1\)

Step-by-Step Solution

Answer

The correct answer is option B: I and III only.

Step 1: Evaluate the given sum

The provided sum is given by \(10+20+30+40\), which adds up to \(100\).

Step 2: Evaluate the given expressions

I. \(\sum_{n=1}^{4} 10 n\) = \(10*1 + 10*2 + 10*3 + 10*4\) = \(10 + 20 + 30 + 40\) = \(100\). II. \(\sum_{n=10}^{40} 10 n\). This expression is incorrect as it would yield much larger values, due to the range of summation starting from 10 going up to 40.III. \(10\left(\sum_{n=1}^{4} n\right)\) = \(10*(1 + 2 + 3 + 4)\) = \(10*10\) = \(100\). This also represents the given sum.

Step 3: Identify the correct options

From step 2, it's clear that options I and III correctly represent the given sum \(10+20+30+40\). Thus, option B: I and III only is the correct answer.

Key Concepts

Arithmetic Series

An arithmetic series is a sequence of numbers in which the difference of any two successive members is a constant. In simpler terms, it's a way of adding numbers that increase (or decrease) steadily. For instance, in the series 10, 20, 30, 40, each number increases by an amount of 10. These numbers form an arithmetic series.
An arithmetic series can also be expressed with a neat formula:

• Sum = number of terms / 2 * (first term + last term)

Using this formula helps in quickly finding the total sum of numbers in such a pattern without adding each number separately. In the example given, the sum can be calculated as:

• Number of terms (n) = 4
• First term = 10
• Last term = 40

Plug these values into the formula:

• Sum = 4/2 * (10 + 40) = 2 * 50 = 100

Thus, the arithmetic series simplifies and speeds up the summation process!

Sigma Notation

Sigma notation is a shorthand way to express the summation of a series of numbers. It uses the Greek letter \( \Sigma \) (sigma) to denote the sum. Instead of writing out every single term, we can use sigma notation to condense the expression.
For our exercise, one expression is \( \sum_{n=1}^{4} 10n \). This translates to adding each term obtained by multiplying 10 with n, where n ranges from 1 to 4.
Some important things about sigma notation:

• The letter at the bottom of the sigma (usually n) indicates the variable used for substitution in the function next to it.
• The numbers above and below the sigma indicate the starting and ending terms, respectively.
• For example, \( \sum_{n=1}^{4} 10n \) expands to 10*1, 10*2, 10*3, and 10*4.

Using sigma notation simplifies representing and calculating long sums, making complex mathematical operations manageable.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They are a way to generalize mathematical problems and can represent both immediate calculations and more abstract ideas. In our exercise, multiple algebraic expressions represent the sum of 10, 20, 30, and 40.
Consider the expression \(10\left(\sum_{n=1}^{4} n\right) \). Here, the expression inside the parentheses \( \sum_{n=1}^{4} n \) means to add the integers 1 through 4. Then, 10 multiplies this sum. Such algebraic expressions are compact and easy to evaluate when broken down step-by-step.
Key Points to Remember:

• Algebraic expressions can include constants (like 10), variables (like n), operations (like addition or multiplication), and may also involve more advanced functions.
• Evaluating algebraic expressions involves substituting numbers for variables and performing arithmetic operations in the correct sequence.

By understanding algebraic expressions, students can break down problems and approach them with a clear, logical method.