Problem 10 - University Calculus: Early Transcendentals

Question

Which formula is not equivalent to the other two? a. \(\sum_{k=1}^{4}(k-1)^{2}\) b. \(\sum_{k=-1}^{3}(k+1)^{2}\) c. \(\sum_{k=-3}^{-1} k^{2}\)

Step-by-Step Solution

Answer

Formula b is not equivalent to the other two.

Step 1: Analyze Formula a

The formula given is \( \sum_{k=1}^{4}(k-1)^{2} \). We need to expand this by evaluating each term for \( k = 1, 2, 3, \text{and } 4 \). \[ (1-1)^2 + (2-1)^2 + (3-1)^2 + (4-1)^2 = 0 + 1 + 4 + 9 = 14 \]

Step 2: Analyze Formula b

The formula given is \( \sum_{k=-1}^{3}(k+1)^{2} \). We need to expand this by evaluating each term for \( k = -1, 0, 1, 2, \text{and } 3 \). \[ (-1+1)^2 + (0+1)^2 + (1+1)^2 + (2+1)^2 + (3+1)^2 = 0 + 1 + 4 + 9 + 16 = 30\]

Step 3: Analyze Formula c

The formula given is \( \sum_{k=-3}^{-1} k^2 \). Calculate each term for \( k = -3, -2, \text{and} -1 \).\[ (-3)^2 + (-2)^2 + (-1)^2 = 9 + 4 + 1 = 14 \]

Step 4: Compare Results

From the calculations, formula a gives \( 14 \), formula b gives \( 30 \), and formula c gives \( 14 \). Formulas a and c result in the same sum, while formula b results in a different sum.

Key Concepts

Equivalent Expressions

Understanding equivalent expressions is crucial in algebra and calculus. In mathematics, equivalent expressions are different expressions that name the same number or represent the same mathematical value. Evaluating equivalent expressions involves substituting variables or constants into an expression and simplifying.
For example, let's consider the expressions evaluated in the exercise:
- Expression a: \( \sum_{k=1}^{4}(k-1)^{2} \)- Expression c: \( \sum_{k=-3}^{-1} k^{2} \)Both of these expressions yield the same mathematical value, 14, even though they look different. The equivalent property lies in their function to produce the same sum when evaluated.
When working with summation notations, it's vital to recognize that different-looking expressions could be equivalent if, after simplification or evaluation, their results are the same. This understanding aids in identifying the correct or equivalent mathematical forms in broader problem-solving scenarios, like comparing or verifying equations.

Arithmetic Sequences

Arithmetic sequences form a series where each term after the first is generated by adding a constant value, known as the common difference, to the previous term. However, the summations in this particular exercise don't directly employ an arithmetic sequence because they involve squared terms, yet understanding periodic sequences assists in understanding the broader context of sequences.
In essence, an arithmetic sequence can be expressed as follows:- Let the first term be \( a \) and the common difference be \( d \).- The sequence is \( a, a+d, a+2d, a+3d, \ldots \)Recognizing patterns in sequences enables students to predict values and sums without calculating each term individually. Furthermore, comprehending different kinds of sequences, including non-arithmetic, ensures better problem-solving techniques across various mathematical topics.

Series Evaluation

Series evaluation refers to calculating the sum of the elements of a sequence. This concept is essential while dealing with finite or infinite series in mathematics. In the context of the given exercise, evaluating each formula involves finding the sum using the summation sign, \( \sum \).
To properly evaluate a series:1. Identify the range of values for the variable of summation (in this case, \( k \)).2. Substitute each term within this range into the expression.3. Calculate each term and then find their sum.
For instance, in Formula a, each term \((k-1)^{2}\) is computed as:\[ (1-1)^2 + (2-1)^2 + (3-1)^2 + (4-1)^2 = 0 + 1 + 4 + 9 = 14 \] Whereas, in Formula b, \((k+1)^{2}\) computes as:\[ (-1+1)^2 + (0+1)^2 + (1+1)^2 + (2+1)^2 + (3+1)^2 = 0 + 1 + 4 + 9 + 16 = 30 \]
Evaluating series requires meticulous attention to detail, especially ensuring that each term is accurately calculated and sums are done precisely. This competence is tremendously beneficial in both academic and real-world applications like finance, engineering, and computer science.