Problem 70 - Algebra for College Students

Question

\sum_{i=4}^{7}\left(3 i^{2}-2\right) \quad 370

Step-by-Step Solution

Answer

The result of the summation is 370.

Step 1: Understand the Summation

The notation \( \sum_{i=4}^{7}(3i^2-2) \) indicates that we need to evaluate the expression \( 3i^2-2 \) for each integer \( i \) from 4 to 7, inclusive, and then add these values together.

Step 2: Calculate Each Term of the Summation

Calculate \( 3i^2-2 \) for each value of \( i \) from 4 to 7:- For \( i=4 \): \[ 3(4^2) - 2 = 3(16) - 2 = 48 - 2 = 46 \]- For \( i=5 \): \[ 3(5^2) - 2 = 3(25) - 2 = 75 - 2 = 73 \]- For \( i=6 \): \[ 3(6^2) - 2 = 3(36) - 2 = 108 - 2 = 106 \]- For \( i=7 \): \[ 3(7^2) - 2 = 3(49) - 2 = 147 - 2 = 145 \]

Step 3: Sum the Calculated Terms

Add the results from each of the calculations:\[ 46 + 73 + 106 + 145 = 370 \]

Step 4: Verify the Given Result

The calculation from Step 3 shows that the sum of these terms is 370, which matches the given answer in the exercise. Therefore, our solution is verified.

Key Concepts

Step-by-Step Calculation

Breaking down mathematical problems into clear, manageable steps is crucial for understanding and solving them effectively. The step-by-step approach allows us to meticulously address each component of a problem.
Firstly, identify the requirements of the problem. For instance, in our exercise, we recognize the need to calculate values for a series of numbers.
Next, compute each value individually, ensuring precision in calculations at every step.
Finally, compile these results to form the final solution. By following this structured method, we minimize errors and clearly understand the logic behind each calculation. This approach is particularly helpful in reducing the complexity of problems while enhancing comprehension.

Polynomial Evaluation

Polynomial evaluation involves determining the value of a polynomial expression for given values of its variables. In this exercise, the polynomial we focus on is in the form of \(3i^2-2\).
To evaluate a polynomial for a specific value, substitute the value into the polynomial expression. For example, substituting \(i=4\) into \(3i^2-2\) results in \(46\).
Repeat this process for other values of \(i\) as needed, as demonstrated in the solution where \(i\) ranges from 4 to 7.
Evaluating polynomials accurately is essential in understanding the behavior of their graphs and finding solutions to equations derived from them.

Summation Notation

Summation notation, represented by the sigma symbol \(\sum\), is a compact way of writing the sum of a sequence of terms. It specifies the starting and ending indices, here from 4 to 7, and the expression to be summed, \(3i^2-2\).
This notation is widely used in mathematics to simplify repeated addition, especially when dealing with sequences or series.
The upper and lower limits indicate where the summation begins and ends, allowing us to calculate multiple terms efficiently.
This structured form is essential in higher mathematics, including calculus and statistics, making it a powerful tool for representing long sums succinctly.

Arithmetic Sequence

Although the given problem involves a polynomial evaluation, it's worth noting the concept of arithmetic sequences to contrast them. An arithmetic sequence is a sequence of numbers in which the difference of any two successive members is a constant.
For instance, in an arithmetic sequence such as 3, 6, 9, each term after the first is obtained by adding a fixed constant, in this case, 3, to the previous term.
Understanding this type of sequence helps in identifying patterns and can aid in solving more complex series that may include polynomial expressions or be expressed in summation notation similar to arithmetic progressions.