Question

Are the expressions in Problems equivalent? \(\sum_{i=0}^{5}(3 i-2)\) and \(\sum_{i=1}^{6}(3 i-5)\)

Short answer

Answer: Yes, both expressions are equivalent.

Step-by-step solution

Write out the terms for both summations.

For the first summation, \(\sum_{i=0}^{5}(3i-2)\), we have the following terms: \[ (3 \cdot 0 - 2) + (3 \cdot 1 - 2) + (3 \cdot 2 - 2) + (3 \cdot 3 - 2) + (3 \cdot 4 - 2) + (3 \cdot 5 - 2) \] For the second summation, \(\sum_{i=1}^{6}(3i-5)\), we have the following terms: \[ (3 \cdot 1 - 5) + (3 \cdot 2 - 5) + (3 \cdot 3 - 5) + (3 \cdot 4 - 5) + (3 \cdot 5 - 5) + (3 \cdot 6 - 5) \]

Simplify each term in both summations.

First, we will simplify the terms in the first summation: \[(-2) + (1) + (4) + (7) + (10) + (13)\] For the second summation, we will also simplify the terms: \[(-2) + (1) + (4) + (7) + (10) + (13) \]

Add the terms and compare the results for both summations.

Now, we will add the terms in each summation. For the first summation, we have: \[(-2) + (1) + (4) + (7) + (10) + (13) = 33 \] For the second summation, we also have: \[(-2) + (1) + (4) + (7) + (10) + (13) = 33 \] Since both summations result in the same value, 33, we can conclude that the expressions in both problems are equivalent.

Key concepts

Summation Notation

Summation notation is a convenient way to represent the addition of a sequence of numbers in a compact form. It's often used in mathematics to express long sums without writing out each term. In summation notation, the Greek letter Sigma (\( \Sigma\)) is used. It specifies the index of summation, which usually begins at a specific initial value and ends at a final value. In our example, we have:

• \( \sum_{i=0}^{5}(3i-2) \) indicating a sum starting at index \( i=0 \) and ending at \( i=5 \).
• \( \sum_{i=1}^{6}(3i-5) \) with the sum beginning at \( i=1 \) and concluding at \( i=6 \).

Each term in the sequence is generated by substituting successive values of \( i\) into the expression inside the summation. This method simplifies the writing of long addition processes and helps in understanding pattern recognition in sequences.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves combining like terms to make the expression easier to work with. In our exercise, we simplified the terms inside the summation by performing the arithmetic indicated.

• Take \( 3i-2 \), substitute each value from \( i=0 \) to \( i=5 \), and compute the result.
• Similarly, substitute each value from \( i=1 \) to \( i=6 \) in \( 3i-5 \), and calculate the outcome.

By handling the terms separately, each term becomes a number. Then all results are summed together. The simplification process aids in quickly identifying patterns and finding equivalent expressions.

Algebraic Equivalence

Algebraic equivalence tells us when two expressions represent the same amount or quantity, even if they appear different at first glance. For expressions to be equivalent, they must yield the same result when all terms are fully simplified and computed.

• Both expressions \( \sum_{i=0}^{5}(3i-2) \) and \( \sum_{i=1}^{6}(3i-5) \) were simplified to the same list of terms: \((-2), 1, 4, 7, 10, 13\).
• The final sum for both is 33, proving that each summation although appearing in slightly different forms, is algebraically equivalent.

Understanding algebraic equivalence can be particularly useful in verifying solutions and ensuring there are no mistakes in algebraic manipulations.