Problem 32 - Calculus for Biology and Medicine

Question

Find an expression for \(a_{n}\) on the basis of the values of \(a_{0}, a_{1}, a_{2}, \ldots .\) $$ 2,-4,6,-8,10, \ldots $$

Step-by-Step Solution

Answer

The expression for the nth term is \( a_{n} = (-1)^{n-1} \times 2n \).

Step 1: Identify the pattern

The sequence provided is 2, -4, 6, -8, 10, ... . We start by identifying whether it's arithmetic, geometric, or another type of sequence. By examining the sequence, we can see that the terms alternate in sign and increase in absolute value.

Step 2: Find the absolute value pattern

Identify that the absolute values are in an arithmetic sequence: 2, 4, 6, 8, 10, which have a common difference of 2. This indicates that the absolute value of each term can be represented as 2n, where n starts from 1.

Step 3: Determine the sign pattern

Next, identify how the signs of the sequence alternate. The sequence starts with a positive term, then alternates. To describe this sign change, observe that even-numbered terms are negative and odd-numbered terms are positive. The general sign for term n is (-1)^{n-1}.

Step 4: Formulate the general term

Using the patterns identified, the nth term of the sequence can be described by including both the absolute value and sign pattern: a_{n} = (-1)^{n-1} imes 2n.

Key Concepts

Arithmetic Sequence

In mathematics, an arithmetic sequence is a sequence of numbers with a constant difference between consecutive terms. This difference is known as the common difference. You can easily recognize an arithmetic sequence because this consistent difference is added or subtracted from each term to get to the next one. For example, in the sequence 2, 5, 8, 11, each term increases by 3, which is the common difference.

• The formula to find the nth term, \(a_n\), of an arithmetic sequence is: \(a_n = a_1 + (n-1) \cdot d\), where \(a_1\) is the first term and \(d\) is the common difference.
• Using this formula, each term can be calculated by knowing the first term and the difference.

In the original exercise sequence, only the absolute value of the terms constitutes an arithmetic sequence with a constant difference of 2, giving us absolute values like 2, 4, 6, 8, 10. Understanding this part of the sequence is crucial before dealing with additional alternations like sign changes.

Geometric Sequence

A geometric sequence, unlike an arithmetic sequence, involves a constant ratio between subsequent terms. This means that to get from one term to the next, you multiply by a specific factor. For instance, in the sequence 3, 6, 12, 24, the ratio between terms is 2.

• The formula for the nth term of a geometric sequence is \(a_n = a_1 \cdot r^{n-1}\), where \(a_1\) is the first term and \(r\) is the common ratio.
• This approach is useful when terms grow or shrink exponentially rather than by a constant addition or subtraction.

However, the original exercise does not follow a geometric sequence pattern. The values do not multiply by a factor; rather, they alternate signs with a constant arithmetic increase in absolute value. This distinction is vital to avoid confusion when identifying sequence types.

Sign Pattern

A sign pattern in a sequence is an organized way in which the signs (positive or negative) of terms change. In sequences where terms alternate in sign, identifying this pattern becomes key. For the sequence given: 2, -4, 6, -8, 10, the pattern starts with a positive and flips signs each time.

• One way to represent alternating signs is through expressions such as \((-1)^{n-1}\). This notation helps describe sequences where signs alternate predictably.

Here, since the first term is positive, you see that odd-numbered terms are positive and even-numbered ones remain negative. This alternating pattern couples with the arithmetic sequence of absolute values to fully describe the sequence. Such a combination of arithmetic sequences for values and a sign pattern results in the general formula combining these elements to express the nth term.