Problem 41 - Algebra for College Students

Question

$$ \sum_{i=2}^{5}(-3)^{i+1} 540 $$

Step-by-Step Solution

Answer

The result of the summation is 291600.

Step 1: Understanding the Summation

The problem presents a summation: \( \sum_{i=2}^{5}(-3)^{i+1} \times 540 \). This means we need to evaluate the expression \((-3)^{i+1} \times 540\) for each integer value of \(i\) from 2 to 5 and then sum all the results together.

Step 2: Compute Each Term in the Summation

1. For \(i=2\), compute \((-3)^{3} \times 540\): \[ (-3)^3 = -27, \quad (-27) \times 540 = -14580 \]2. For \(i=3\), compute \((-3)^{4} \times 540\): \[ (-3)^4 = 81, \quad 81 \times 540 = 43740 \]3. For \(i=4\), compute \((-3)^{5} \times 540\): \[ (-3)^5 = -243, \quad (-243) \times 540 = -131220 \]4. For \(i=5\), compute \((-3)^{6} \times 540\): \[ (-3)^6 = 729, \quad 729 \times 540 = 393660 \]

Step 3: Sum the Computed Values

Add all the computed terms from the previous step:\[(-14580) + 43740 + (-131220) + 393660\]Calculate the sum step by step:\[-14580 + 43740 = 29160\]\[29160 - 131220 = -102060\]\[-102060 + 393660 = 291600\]

Step 4: Conclusion: Final Result

The final result of the summation is 291600. This value is calculated by computing and summing the individual terms of the expression for \(i\) ranging from 2 to 5.

Key Concepts

Exponential Function

An exponential function is a mathematical expression where a number, known as the base, is raised to a power, which is the exponent. In our problem, the base is (-3) and the exponent depend on the variable i. Exponential functions grow or decay at increasingly rapid rates. For instance:

• When the exponent is positive, the function grows because the number is multiplied by itself multiple times.
• When the base is negative, the function typically alternates between positive and negative values depending on whether the exponent is odd or even. This property is critical when analyzing functions like pow((-3),i+1), which toggles sign as i changes.

Understanding these properties helps solve and predict the outcome of equations that involve exponential functions, especially since small changes in the exponent can lead to significant changes in the result.

Arithmetic Operations

Arithmetic operations such as addition, subtraction, multiplication, and division are fundamental to evaluating mathematical expressions. In our example, after computing (-3) raised to various powers, each result is multiplied by 540. This multiplication is crucial, as it scales the exponential component:

• Multiplying exponential results allows scaling of values to fit specific contexts, like adjusting a graph's slope or resizing data.
• All these scaled results are summed incrementally to arrive at a final total. Understanding the neat order of operations—PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction)—ensures accuracy.

Keeping track of signs when performing operations is pivotal, especially given how positive and negative signs interact during addition or subtraction. Understanding these arithmetic principles assists in correctly evaluating complex expressions like the summation in our exercise.

Series Calculation

Calculating a series involves evaluating and summing a sequence of numbers. In this exercise, we are tasked with summing specific terms generated through the expression

First, determine the terms in the series: here, the expression changes with each integer value from 2 to 5.

After determining these terms, compute their individual values as shown in preceding steps.

Finally, add these values together to find the series' total value.

This methodical breakdown assists in understanding and calculating series with multiple terms, laying a solid foundation for tackling more advanced topics.

Mathematical Notation

Mathematical notation provides a systematic way to represent numbers, functions, and operations in a concise and universally understood format. In this problem, we use the \( \sum \) symbol, which signifies a summation.

• The subscript and superscript indicate the range of values over which the summation occurs. For instance, \(i=2\) to \(5\) sets the iteration bounds in our problem.
• Inside the summation symbol, the expression indicates the variable terms being summed, a key feature of notation that guides computations.

Familiarity with mathematical symbols and their meanings is essential for interpreting and executing equations accurately. Learning this language allows one to decipher and translate mathematical queries into actionable steps, enhancing problem-solving skills.