Problem 93 - Precalculus with Limits

Question

In Exercises 85-96, find the sum. \( \displaystyle \sum_{k=2}^{5} (k + 1)^2 (k - 3) \)

Step-by-Step Solution

Answer

The sum is 88

Step 1: Evaluate each term individually

Using the formula \((k + 1)^2 (k - 3)\), replace \(k\) with each integer from 2 to 5. This gives the expressions: \[ (2 + 1)^2*(2 - 3)\], \[ (3 + 1)^2*(3 - 3)\], \[ (4 + 1)^2*(4 - 3)\], \[ (5 + 1)^2*(5 - 3)\]

Step 2: Simplify each expression

Simplify each expression calculated in Step 1: \[ (2 + 1)^2*(2 - 3) = 9*(-1) = -9\] \[ (3 + 1)^2*(3 - 3) = 16*0 = 0\] \[ (4 + 1)^2*(4 - 3) = 25*1 = 25\] \[ (5 + 1)^2*(5 - 3) = 36*2 = 72\]

Step 3: Add the results

The last step is to sum all the results together. Adding -9, 0, 25, 72, we get: -9 + 0 + 25 + 72 = 88

Key Concepts

Sequences and Series

When we encounter the expression \( \sum_{k=2}^{5} (k + 1)^2 (k - 3) \), we're diving into the mathematical concept of sequences and series. A **sequence** is essentially a list of numbers arranged in a specific order. In our example, this sequence comprises of terms that result from substituting each integer between 2 and 5 into the expression \( (k+1)^2(k-3) \). Each number is derived through this formula by plugging in different values for \( k \).

On the other hand, a **series** refers to the act of adding these numbers together. In our exercise, once we have determined the sequence, we sum the results to find the total. The notation \( \sum \) indicates a summation across our designated values of \( k \). Understanding sequences and series helps us see patterns and sum numbers efficiently in both mathematical and real-world problems.

Algebraic Expressions

Algebraic expressions are the building blocks of algebra. They consist of numbers, variables, and arithmetic operations combined in a meaningful way. In our example, the expression \( (k + 1)^2 (k - 3) \) is an algebraic expression. It combines the variable \( k \) with constants (1 and -3) and operations (addition, subtraction, and multiplication).

The expression can be broken into two parts:

• Inner Operations: First, we handle the arithmetic inside the parentheses; adding 1 to \( k \) and subtracting 3 from \( k \).
• Outer Operations: Next, we square the result of \( (k + 1) \) and then multiply it by \( (k - 3) \).

Algebraic expressions like these allow us to represent relationships in mathematics that involve changing numbers or variables. Mastering them is essential for solving algebraic equations, simplifying expressions, and understanding mathematical models.

Arithmetic Operations

Arithmetic operations are the foundation of mathematics, allowing us to perform calculations and solve problems. In our example, the main operations include addition, subtraction, multiplication, and the use of exponents (powers). Let's break them down:

• Addition and Subtraction: Inside the expression \((k + 1)^2 (k - 3)\), we're first adjusting \( k \) by adding or subtracting constants (1 and 3).
• Multiplication: Once calculated, the results of \( k+1 \) and \( k-3 \) are then multiplied together after considering the exponent effect.
• Exponentiation: Squaring \( (k + 1) \) means multiplying it by itself, which is an example of applying exponents.

Through these steps, we transform \( k \) according to our equation and then sum all end results as instructed in the series. Proficiency with arithmetic operations not only helps in evaluating algebraic expressions but also strengthens one's general mathematical problem-solving skills.