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

\(a_n = (-1)^n (n + 1)\).

Step 1: Look for a pattern

Examine the given sequence: \(-1, 2, -3, 4, -5, \ldots\). Notice that the sequence alternates between negative and positive numbers, and the absolute value of each term increases linearly, matching the sequence of natural numbers: 1, 2, 3, 4, 5,...

Step 2: Determine the sign pattern

The sequence has a sign pattern: the signs alternate starting with negative for \(a_0\), positive for \(a_1\), negative for \(a_2\), etc. This alternating sign can be represented by \((-1)^n\), since \((-1)^n\) results in -1 for even \(n\) and 1 for odd \(n\).

Step 3: Determine the magnitude pattern

The magnitude of each term increases as the sequence of natural numbers: 1, 2, 3, 4, 5,.... Thus, for each \(a_n\), the magnitude matches \(n + 1\).

Step 4: Combine sign and magnitude to form the general formula

Combine the results from Steps 2 and 3. The general term for the sequence is given by the product of the alternating sign and the magnitude sequence: \(a_n = (-1)^n (n + 1)\). This formula will generate the sequence: \(-1\) for \(n=0\), \(2\) for \(n=1\), \(-3\) for \(n=2\), etc.

Key Concepts

Alternating Sequence

An alternating sequence is a sequence in which the signs of the terms alternate between positive and negative. In the original exercise, the sequence -1, 2, -3, 4, -5, ... illustrates this concept perfectly. Here we see positive numbers are followed by negative numbers and vice versa in a regular pattern. This alternating pattern is fundamental because it shows how the sequence can toggle signs, creating a distinctive oscillating pattern that can often be described mathematically. Understanding this alternating pattern allows us to use mathematical tools such as (-1)^n to succinctly represent the sequence's alternating property. Where the term is negative, like for even indexed terms, (-1)^n equals -1, and for odd indexed terms, it equals 1. This mathematical representation is crucial for finding a formula for the general term of the sequence.

Sequence Pattern

The pattern within a sequence refers to the consistent, predictable rule that defines how terms are arranged. Recognizing the sequence pattern is key to understanding both the sign and the magnitude of terms in sequences such as -1, 2, -3, 4, -5, ... In our case, the absolute value of each term in the sequence mirrors the sequence of natural numbers:

• For a_0 , the value is 1 ,
• For a_1 , the value is 2 ,
• For a_2 , the value is 3 ,

This pattern shows that each term's magnitude can be represented as n+1 for the nth term. It's this predictability that helps configure the general term formula. By understanding this, students can decipher not just specific sequences, but any sequences where a similar linear pattern exists.

General Term Formula

The general term formula in a sequence provides a way to calculate any term in the sequence without having to list all preceding terms. It essentially acts as an efficient shortcut.In the provided exercise, our task is to combine the insights from recognizing the alternating and magnitude patterns. This results in the general term formula:\[a_n = (-1)^n (n + 1)\]Here's how it works:

• The (-1)^n defines the alternating sign.
• The (n + 1) describes the sequence's magnitude, aligned with natural numbers.

So when you're given any specific nd, you can plug it into this formula to get a_n. For example, for n = 3, the formula gives a_3 = 4, confirming consistency with the pattern. This ability to calculate terms helps one move beyond simple pattern recognition to quantitative determination.