Problem 8 - Thomas Calculus

Question

Which of the following express \(1-2+4-8+16-32\) in sigma notation? $$\text { a. } sum_{k=1}^{6}(-2)^{k-1} \quad \text { b. } \sum_{k=0}^{5}(-1)^{k} 2^{k} \quad \text { c. } \sum_{k=-2}^{3}(-1)^{k+1} 2^{k+2}$$

Step-by-Step Solution

Answer

Options a and b express the sequence in sigma notation.

Step 1: Identify the Pattern

Observe the alternating sequence \(1, -2, 4, -8, 16, -32\). Notice it changes sign between positive and negative and each number is a power of 2. This suggests the sequence can be defined with \((-1)^k\) for alternating signs and \(2^k\) for powers of 2.

Step 2: Examine Option (a)

The expression \(\sum_{k=1}^{6}(-2)^{k-1}\) expands to terms like \(1, -2, 4, -8, 16, -32\).This matches the given sequence because \((-2)^{k-1} = (-1)^{k-1} \cdot 2^{k-1}\) follows the same pattern of powers and alternating signs.

Step 3: Examine Option (b)

The expression \(\sum_{k=0}^{5}(-1)^{k} 2^{k}\) expands to \(1, -2, 4, -8, 16, -32\). It uses alternating signs, \((-1)^k\), and powers of 2, \(2^k\), starting from \(k=0\). This also matches the original sequence.

Step 4: Examine Option (c)

The expression \(\sum_{k=-2}^{3}(-1)^{k+1} 2^{k+2}\) expands with power adjustments. It includes terms \(\cdots , -32, 64\), which differ from our sequence. Hence, this option does not match the original sequence.

Step 5: Conclusion

Both \(\text{a. } \sum_{k=1}^{6}(-2)^{k-1}\) and \(\text{b. } \sum_{k=0}^{5}(-1)^{k} 2^{k}\) express the sequence \(1-2+4-8+16-32\) in sigma notation. Option c does not.

Key Concepts

Alternating Series

An alternating series is a sequence of numbers where the signs change between positive and negative consecutively. In the context of the given exercise, the sequence is expressed as \(1, -2, 4, -8, 16, -32\). This pattern of alternating signs can be represented using the mathematical notation \((-1)^k\). With each change in the index \(k\), the sign flips due to the nature of raising \(-1\) to sequential powers.

• When \(k\) is odd, \((-1)^k\) results in -1, giving a negative term.
• When \(k\) is even, \((-1)^k\) results in 1, giving a positive term.

This alternating behavior is essential when utilizing sigma notation to define sequences like the one in the exercise, ensuring each term alternates correctly between positive and negative.

Power of 2

Within the sequence \(1, -2, 4, -8, 16, -32\), each absolute value can be described by powers of 2. This is evident as each term can be written as \(2^k\) where \(k\) adjusts according to the term’s position in the sequence. Powers of 2 are pivotal in sequences where growth between numbers is geometric.

• For example, \(4\) is \(2^2\), showing that as each term evolves, it multiplies by 2 from the previous one.
• The representation \(2^k\) is crucial to capture the exponential growth seen in sequences of powers of 2.

Understanding powers of 2 allows you to correctly form the sequence using mathematical expressions like those given in sigma notation. This method makes it easy to specify terms systematically, as it leverages the exponential pattern of powers.

Sequence Patterns

Recognizing sequence patterns involves identifying the regularity within a list of numbers. In mathematical problems, identifying sequence patterns is fundamental to describe the structure concisely using sigma notation. The sequence \(1, -2, 4, -8, 16, -32\) demonstrates a dual pattern: alternating signs and powers of 2.

• The alternating sign pattern is managed by adding \((-1)^k\), resulting in the expected positive and negative flips between terms.
• The pattern of powers of 2 is captured by using \(2^k\), which reflects how each term is scaled geometrically from a base value.

Identifying these two core patterns makes it possible to express the sequence accurately using a formula. This facilitates a greater understanding of the underlying maths through concise, repeatable procedures like those found in sigma notation.