Problem 31 - Calculus for Biology and Medicine

Question

Find an expression for \(a_{n}\) on the basis of the values of \(a_{0}, a_{1}, a_{2}, \ldots .\) $$ -1,2,-3,4,-5, \ldots $$

Step-by-Step Solution

Answer

The expression for \(a_n\) is \((-1)^n (n+1)\).

Step 1: Identify the Pattern

Look at the given sequence: \(-1, 2, -3, 4, -5, \ldots\). Observe the signs and the values to identify a pattern: the signs alternate between negative and positive, and the absolute values increase by 1 each time.

Step 2: Establish Sign Alternation

Since the sequence alternates signs, use \((-1)^n\) to describe the alternating sign behavior. This expression returns \(-1\) for odd \(n\) and \(+1\) for even \(n\).

Step 3: Determine the Absolute Values

The absolute values of the sequence follow the pattern 1, 2, 3, 4, 5, ..., which can be represented as \(n+1\) for \(a_n\).

Step 4: Combine Both Patterns

Combine the alternation and absolute value patterns to form the full expression for \(a_n\). This is done by multiplying the absolute pattern by the sign alternation: \(a_n = (-1)^n (n+1)\).

Key Concepts

Alternating Sequences

An alternating sequence is a special type of sequence where the signs of the terms oscillate between positive and negative. In the sequence provided in the exercise

• -1, 2, -3, 4, -5, ...

the pattern of alternating signs can be clearly observed. To describe such a sequence mathematically, we use the term

• 
  (-1)^n.

This notation is powerful because:

• It gives -1 when "n" is odd, indicating a negative term.
• It gives +1 when "n" is even, indicating a positive term.

This concept of alternating sequences is common in mathematics because it can help denote or model situations where values swap directions or characteristics systematically.

Recurrence Relations

Recurrence relations are equations that express each term of a sequence in relation to its predecessor(s). In the context of our given sequence

• -1, 2, -3, 4, -5, ...

we don't have an explicit recurrence relation since the sequence is directly described by alternating signs combined with incremental absolute values. However, if you wanted to express the relationship and form it as a recurrence:

• You might observe that each term relies on knowing its position, "n", within the sequence.
• Every subsequent term's absolute value increases by exactly 1.

Recurrence relations can enhance your understanding of sequences when direct formulas aren’t available. They provide another way to think about sequences by expressing how terms grow or change based on previous terms.

Mathematical Induction

Mathematical induction is a method of mathematical proof typically used to establish that a given statement holds true for all natural numbers. It has two basic steps:

• Base Case: Prove the statement is true for the initial value, usually n = 0 or n = 1.
• Inductive Step: Assume the statement holds for some arbitrary natural number k, and then prove it for k+1.

In the context of the sequence given, you could use induction to show that the formula for the sequence

• a_n = (-1)^n (n+1)

holds for all integers n.

• Start by proving it for n = 0: clearly, a_0 = -1, which fits the formula perfectly.
• Then, assume a_k = (-1)^k (k+1) is correct, and prove a_{k+1} = (-1)^{k+1} (k+2) fits based on this assumption.

Mathematical induction is a reliable technique for proving formulas that apply over sequences and is particularly useful for affirming patterns that extend indefinitely.