Problem 2 - Active Calculus

Question

Find a formula for \(s_{n}, n \geq 1\) for the sequence \(-3,5,-7,9,-11 \ldots\)

Step-by-Step Solution

Answer

The formula for the Sequence is s_{n}= (-1)^{n+1}(2n+1).

Step 1 - Identify the Pattern

Examine the sequence: -3, 5, -7, 9, -11, Notice the alternating signs and determine how the absolute values progress: 3, 5, 7, 9, 11. The absolute values increment by 2 each time.

Step 2 - Establish the Rule for the Absolute Value

Since the absolute values increase by 2, the terms can be written as 2n + 1 or similar. Test this to ensure it matches the absolute values. For instance, the first term is |2(1)-1| = 3, the second term is |2(2)+1| = 5, and so forth.

Step 3 - Determine the Sign Pattern

The signs alternate: The 1st term is negative, the 2nd term positive: Observe that (-1)^{n+1} provides the appropriate pattern.

Step 4 - Combine the Formulas

Combine the steps to obtain the formula factoring in signs: s_{n} = (-1)^{n+1}(2n + 1). This formula ensures the correct alternation in signs while accommodating the sequence structure.

Key Concepts

Pattern Recognition

Recognizing patterns is a fundamental skill in solving sequences. Patterns help us see the structure of a sequence and predict future terms. In the given sequence \[-3, 5, -7, 9, -11 \dots\], the first observation is the **alternation of signs**.
The sequence alternates between negative and positive values.
For example, terms go from -3 (negative) to 5 (positive), then back to -7 (negative).
Additionally, the sequence reveals a pattern in the absolute values of each term. These are \[3, 5, 7, 9, 11 \dots\], incrementing by 2.
Recognizing these patterns is the first step to building a formula that fits all terms of the sequence.

Sign Alternation

Sign alternation is a key feature in many sequences, including our current example. To capture this alternation:

• Notice how the sequence alternates from negative to positive.
• The mathematical expression \[(-1)^{n+1}\]fits this pattern perfectly.
  When n is 1, it evaluates to \[(-1)^{2} = 1\]making the term positive.
  When n is 2, it evaluates to \[(-1)^{3} = -1\], making the term negative.

This consistent switch between terms ensures the formula encapsulates the alternating nature of the sequence.
By incorporating this sign pattern into the formula, all terms will correctly alternate.

Formula Derivation

Deriving a formula involves combining the detected patterns and expressions into a cohesive whole. Here's how:

• Start by realizing the absolute values follow \[|2n - 1|\], which was found through **Pattern Recognition**.
• Add the **Sign Alternation** component: \[(-1)^{n+1}\].

Combining these aspects, we derive the general term's formula:
\[s_n = (-1)^{n+1}(2n - 1)\].
This formula aligns with each term in the sequence, correctly interchanging signs while matching the absolute values.

Sequence Analysis

Analyzing a sequence goes beyond just finding the next term. It involves:

• Deeply inspecting the sequence's growth rate or pattern.
• Deriving relationships between terms.
  For instance, in our sequence: \[-3, 5, -7, 9, -11\dots\] we identify a common difference of 2 in the absolute values.

Effective analysis helps in understanding the sequence structure, further verifying if our derived formula \[s_n = (-1)^{n+1}(2n - 1)\] holds true for different values of n.
Plugging various n values (e.g., 1, 2, 3) ensures the formula generates terms that match the initial sequence, offering complete verification.