Problem 95 - College Algebra

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. It makes a difference whether or not I use parentheses around the expression following the summation symbol, because the value of \(\sum_{i=1}^{5}(i+7)\) is \(92,\) but the value of \(\sum_{i=1}^{8} i+7\) is $43 .

Step-by-Step Solution

Answer

The statement makes sense, because parentheses significantly impact the interpretation of mathematical expressions, especially in the case of summation. Their presence or absence dictates whether an operation (here, addition by 7) applies to each term being summed or is performed after the summation. Thus, \(\sum_{i=1}^{5}(i+7) = 50\) and \(\sum_{i=1}^{8} i + 7 = 43\), demonstrating the effect of parentheses.

Step 1: Analyze the First Summation

The summation \(\sum_{i=1}^{5}(i+7)\) means you add the result of \(i+7\) where \(i\) ranges from 1 to 5. The parentheses make sure that 7 is added to \(i\) before summing all terms up.

Step 2: Calculate the First Summation

Calculate, following the summation and parentheses rule: = (1+7) + (2+7) + (3+7) + (4+7) + (5+7) = 8+9+10+11+12 = 50.

Step 3: Analyze the Second Summation

In the second summation \(\sum_{i=1}^{8} i+7\), there are no parentheses which means 7 is not part of the operation under the summation. This expression implies adding \(i\) from 1 to 8, then adding 7 to the total sum.

Step 4: Calculate the Second Summation

Calculate following the summation and lack of parentheses: = 1+2+3+4+5+6+7+8 +7= 36+7=43.

Key Concepts

Parentheses in Mathematics

In mathematics, parentheses play a crucial role, especially when dealing with expressions like summations. Parentheses determine the order of operations, ensuring certain calculations happen first. Consider them like a checklist that ensures that operations inside them are prioritized before anything else.

For instance:

• In the expression \(\sum_{i=1}^{5}(i+7)\), the parentheses mean that you should add 7 to each value of \(i\) before summing the results.
• When calculating without parentheses, as in \(\sum_{i=1}^{8} i+7\), standard order applies; you perform the summation first, then add 7 to the final result.

Getting the order wrong can completely change the outcome of the calculation. As shown, the same numbers can lead to different results when parentheses are used or omitted.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operators like addition and multiplication. They form the building blocks of algebra and are essential to solving problems.

These expressions become even more critical when using summation notation since you have to repeatedly apply operations over a range of values.

• Consider \(i+7\): This tells you to add 7 to each value of \(i\).
• Expression \(\sum_{i=1}^{5}(i+7)\) signifies evaluating this expression for each \(i\) from 1 to 5.
• However, \(\sum_{i=1}^{8} i+7\) presents a distinct expression where only \(i\), ranging from 1 to 8, is summed, and then 7 is added.

Recognizing the structure of algebraic expressions helps in knowing how to process them correctly, ensuring accurate calculations.

Mathematical Reasoning

Mathematical reasoning is about understanding why a mathematical statement or solution is true or false. It involves a logical thought process to evaluate and solve problems effectively. This reasoning is exceptionally crucial when interpreting and solving problems with summation notation.

To achieve this, analyze both what the expression means and the implications of changes to it. This ensures a deepened understanding of its outcome.

• As in \(\sum_{i=1}^{5}(i+7)\), being mindful that the terms are grouped within parentheses leads to rearranging the order of addition.
• In contrast, for \(\sum_{i=1}^{8} i+7\), reason determines the distribution of arithmetic tasks across the expression, impacting the final value.

Having strong mathematical reasoning abilities allows for verifying whether steps and calculations make sense, and if the solution is correct.