Question

Are the expressions in Problems equivalent? \(\sum_{i=0}^{5} 5\) and 30

Short answer

Answer: Yes, the expressions are equivalent.

Step-by-step solution

Understand the first expression

The first expression uses the summation notation, which represents the sum of a sequence. It is given by \(\sum_{i=0}^{5} 5\). The formula means that we will add the number 5, six times (i = 0, 1, 2, 3, 4, and 5).

Calculate the sum of the first expression

To calculate the sum, we will add the number 5 for each i-value in the given range. So, the sum will be: \(5 + 5 + 5 + 5 + 5 + 5 = 30\)

Compare the sums

Now that we have found the sum of the first expression to be 30, we can compare it to the second expression, which is also 30. Since both expressions are equal to 30, we can conclude that the expressions are equivalent.

Key concepts

Equivalent Expressions

Equivalent expressions are different mathematical expressions that represent the same quantity or result. In this exercise, the expressions \(\sum_{i=0}^{5} 5\) and 30 are compared to see if they are equivalent. When working with equivalent expressions, it is important to:

• Understand that they can look different but yield the same final value.
• Verify equivalence by simplifying or calculating them, as shown in the solution steps.

In this case, the expression \(\sum_{i=0}^{5} 5\) translates to adding 5 six times, resulting in a total of 30. Because both expressions simplify to the same number, they are indeed considered equivalent.

Summation Formula

Summation notation is a way to write a sequence of numbers to be added together in a compact form. The general form of a summation notation is \(\sum_{i=m}^{n} a_i\), which tells us to add the terms \(a_i\) starting from the integer \(m\) up to \(n\). In our exercise, the summation \(\sum_{i=0}^{5} 5\) indicates that the value 5 is added each time for six iterations, corresponding to the integer values i from 0 to 5.

• It's crucial to examine the formula inside the summation. It tells what consistent number or mathematical expression you are summing.
• The limits (lower bound \(i=0\) and upper bound \(i=5\)) determine the number of terms to add together.

So, understanding this formula guides the process of adding the same number repeatedly, epitomizing the power and simplicity of summation notation.

Sequence Addition

Sequence addition involves adding a series of numbers in an orderly manner. In mathematics, sequences can be created for consistent patterns or rules that define each element. Particularly, sequence addition becomes straightforward when each element in the sequence is the same number, as each step involves adding this repeated element.In the exercise, the expression \(\sum_{i=0}^{5} 5\) provides a perfect example of sequence addition:

• The sequence involves the constant number 5 being added up, reflecting a very predictable pattern.
• By adding 5 for six times, you precisely measure how sequence addition works with repetitive elements within a range.

Therefore, understanding how sequences are built and summed up provides insight into mathematical operations, making it simpler to calculate totals quickly and accurately.