Question

Are the expressions in Problems equivalent? \(\sum_{i=1}^{1} 10\) and 10

Short answer

Answer: Yes, they are equivalent.

Step-by-step solution

Expression 1: Summation expression

We have the summation expression \(\sum_{i=1}^{1} 10\). This means we need to add the value 10 for each integer in the range from 1 to 1 (inclusive).

Expression 2: Number

The second expression is simply the number 10.

Evaluate Expression 1

To evaluate the summation expression, we need to find the sum of the expression for each integer in the specified range. Since the range is just from 1 to 1, it means we only need to evaluate the expression for i = 1. Therefore, the summation expression equals 10.

Comparing the expressions

Now that we've evaluated both expressions, we can compare them. Expression 1 equals 10 and Expression 2 is 10. Since both expressions have the same value, we can conclude that the expressions are equivalent.

Key concepts

Summation Notation

Summation notation is a way to compactly represent the addition of a sequence of numbers. It is denoted by the Greek letter sigma (\( \Sigma \)). The expression \( \sum_{i=1}^{n} a_i \) means we sum the terms \( a_i \) from \( i = 1 \) to \( n \). This notation is particularly useful when dealing with series or repetitive addition of terms. In our exercise, the specific summation notation \( \sum_{i=1}^{1} 10 \) indicates that we only need to add the value 10 one time, because our index \( i \) runs from 1 to 1. The important part to note here is that if the range consists of only one number, like in this case, the sum simplifies to just that single term's value. Understanding this helps in quickly determining the result when the range in a summation is very limited.

Evaluating Expressions

Evaluating expressions involves determining the numerical value of an algebraic expression. This process can involve replacing variables with numbers, simplifying operations, and following mathematical operations such as addition, multiplication, etc.For our summation expression \( \sum_{i=1}^{1} 10 \), evaluating it is simpler than it might first appear. Since we only perform one addition (as the index \( i \) runs from 1 to 1), we simply take the value 10 as the result. Evaluating here is straightforward: when no variables or complex operations are involved, you evaluate by direct substitution or just performing the indicated arithmetic operation. On the other hand, evaluating a standalone number like 10 is even more direct because it simply retains its value. Expression evaluation is a key skill in algebra, allowing you to verify or understand different mathematical expressions and their equivalences.

Equivalent Expressions

Two expressions are said to be equivalent if they simplify to the same value, no matter how they look initially. In algebra, this means that after performing all operations, each expression yields the same result.In the context of our problem, after evaluating the summation expression \( \sum_{i=1}^{1} 10 \) and the number 10, we found that both result in the value 10. This shows they are equivalent because they produce the same outcome. Recognizing equivalent expressions is very useful when solving algebraic equations or when simplifying expressions to compare different forms. It allows us to confirm consistency and correctness in mathematical calculations. Being adept at identifying such equivalences is an essential part of deepening one's mathematical understanding.