Question

Are the expressions in Problems equivalent? \(\sum_{j=1}^{9} 3 a_{j}\) and \(3 a_{1}+3 a_{2}+3 a_{3}+\cdots+3 a_{8}+3 a_{9}\)

Short answer

Answer: Yes, both expressions are equivalent.

Step-by-step solution

Understand the meaning of the summation symbol

The summation symbol is a shorthand way of writing the sum of a series. The expression \(\sum_{j=1}^{9} 3 a_{j}\) represents the sum of 3 times \(a_j\) for each value of \(j\) from 1 to 9. It can be expanded as follows: \(3 a_{1}+3 a_{2}+3 a_{3}+\cdots+3 a_{8}+3 a_{9}\)

Compare the expanded summation expression with the given expression

Now that we have expanded the summation expression, let's compare it with the second given expression: \(3 a_{1}+3 a_{2}+3 a_{3}+\cdots+3 a_{8}+3 a_{9}\) We can see that both expressions are exactly the same.

Conclusion

Since both expressions represent the same series of terms, we can conclude that the expressions are equivalent.

Key concepts

Equivalent Expressions

In algebra, equivalent expressions are expressions that look different but represent the same quantity or value. This may seem tricky, but think of equivalent expressions like two ways to describe the same thing. For example,

• Expression A: \( \sum_{j=1}^{9} 3 a_{j} \)
• Expression B: \( 3 a_{1} + 3 a_{2} + 3 a_{3} + \ldots + 3 a_{9} \)

Although Expression A uses a summation symbol and Expression B writes out each term, they are essentially doing the same mathematical work: adding up all the terms where each term is three times a specific number \(a_j\).
By understanding that different forms can yield the same result, students can develop flexibility in solving problems that appear in different formats.

Expanding Summations

Expanding summations is a process where you take the compact form of a sum, usually denoted with the summation symbol, and write it out in its full form. This helps in visualizing each term in the series.
For the summation \( \sum_{j=1}^{9} 3 a_{j} \), it means you take every integer value of \(j\) from 1 to 9, multiply it by 3, and by the corresponding \(a_j\). When expanded, it appears as:
\[ 3 a_{1} + 3 a_{2} + 3 a_{3} + \dots + 3 a_{9} \]
Each piece of the expanded expression represents a term in the series. Expanding summations is vital because it allows deeper insight into what each term in the expression represents individually.
Moreover, it simplifies understanding of how all terms collectively form the entirety of a mathematical series.

Mathematical Notation

Mathematical notation is like a special language that helps in describing mathematical ideas precisely and concisely. Consider the summation sign \( \sum \): it serves as a shorthand to avoid writing repeated addition over multiple terms.

• \( \sum_{j=1}^{9} 3 a_{j} \) conveys a clearer and more concise message to someone familiar with the notation than listing every term.
• It allows mathematicians and students alike to focus on the equation's structure rather than getting bogged down by repetitive elements.

Adopting this notation sharpens computational strategies and can make complex problems more manageable.
Learning to "read" and interpret these symbols is crucial in advancing math skills, as it creates an efficient way to interpret large sums and facilitates further manipulation in algebraic processes.