Problem 64 - Algebra for College Students

Question

\sum_{i=6}^{7}(4 i-9) \quad 4074

Step-by-Step Solution

Answer

The sum is 34.

Step 1: Setting Up the Summation Formula

We need to evaluate the sum \( \sum_{i=6}^{7}(4i - 9) \). This means we will plug in the integer values starting from \(i = 6\) up to \(i = 7\) into the expression \(4i - 9\).

Step 2: Calculating the Expression for i=6

When \(i = 6\), substitute 6 into the expression \(4i - 9\):\[ 4(6) - 9 = 24 - 9 = 15 \].

Step 3: Calculating the Expression for i=7

When \(i = 7\), substitute 7 into the expression \(4i - 9\):\[ 4(7) - 9 = 28 - 9 = 19 \].

Step 4: Adding the Results

Add the results from when \(i = 6\) and \(i = 7\) to find the sum:\[ 15 + 19 = 34 \].

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operation symbols. They play a critical role in algebra and can represent various mathematical situations. In our original exercise, the expression used is \(4i - 9\). This consists of:

• A coefficient (4) that multiplies the variable \(i\).
• The variable \(i\), which will be replaced with specific values.
• A constant term (-9), which doesn't change as \(i\) changes.

When evaluating algebraic expressions, it's essential to understand how each part interacts as the values of the variable change. For instance, in the expression \(4i - 9\), as \(i\) increases, the result becomes larger because the effect of multiplying by the coefficient 4 outweighs the constant subtraction of 9.

Series

A series in mathematics refers to the sum of terms of a sequence. When discussing a series, we are often looking at adding up elements that follow a specific pattern or rule.
In the context of our exercise, the series is given by the summation notation \(\sum_{i=6}^{7}(4i - 9)\). This notation instructs us to calculate the expression \(4i - 9\) for values of \(i\) starting from 6 and ending at 7, then add up the results.
Series are powerful tools because they allow for the concise expression of potentially many operations. Each term is found by plugging in successive values of the index (in this case, \(i\)), and then those terms are combined. The series is completed once all terms have been evaluated and summed.

Mathematical Notation

Mathematical notation is a system of symbols and signs used to represent numbers and operations concisely. These notations make complex ideas easier to communicate and solve.
A common example is summation notation, seen here as \(\sum_{i=6}^{7}(4i - 9)\). This notation tells us a lot in a small space:

• The symbol \(\sum\) (sigma) indicates that a sum will be performed.
• Items below and above the sigma, \(i=6\) and \(i=7\), denote the starting and ending values of the index \(i\).
• The expression \(4i - 9\) determines what calculated term is being summed at each index value.

Understanding this notation is key to interpreting and solving various problems in mathematics, as it provides a streamlined method for describing complex operations.