Problem 32 - Calculus for Biology and Medicine

Question

Find an expression for \(a_{n}\) on the basis of the values of \(a_{0}, a_{1}, a_{2}, \ldots\) \(9,16,25,36,49\)

Step-by-Step Solution

Answer

The expression for \(a_n\) is \((3+n)^2\).

Step 1 - Identify the Sequence Type

The series given is \(9, 16, 25, 36, 49\). This looks like a sequence of perfect squares. We need to analyze the sequence to derive a formula for \(a_n\).

Step 2 - Analyze Sequence Patterns

Write the sequence terms in terms of their roots: \(3^2 = 9, 4^2 = 16, 5^2 = 25, 6^2 = 36, 7^2 = 49\). This reveals that the terms are squares of consecutive integers starting from 3.

Step 3 - Determine the General Formula

From the pattern, we see the terms can be written as \((3+n)^2\). The number 3 corresponds to the starting integer for squaring. For nth term, where \(n = 0, 1, 2, 3, 4\), we observe that \(a_n = (3+n)^2\).

Step 4 - Verify the Formula

Check the formula with the given terms: for \(n=0\), \(a_0 = (3+0)^2 = 9\); for \(n=1\), \(a_1 = (3+1)^2 = 16\); and similarly for others. All terms match, confirming the formula.

Key Concepts

Perfect Squares

In mathematics, a perfect square is a number that can be expressed as the square of an integer. They hold fascinating properties and appear often in various mathematical contexts.
When we look at our sequence: 9, 16, 25, 36, and 49, we notice that each term can be represented as the square of another number.
Let’s break it down:

• 9 can be written as \(3^2\)
• 16 as \(4^2\)
• 25 as \(5^2\)
• 36 as \(6^2\)
• 49 as \(7^2\)

Thus, in this specific sequence, each term is a perfect square of consecutive numbers starting from 3. This type of identification is crucial because it guides us towards understanding the deeper patterns and subsequent formula derivations for sequences.

Sequence Patterns

Identifying patterns in sequences is key to finding relationships between terms. Patterns allow us to make predictions and understand the rules that govern sequences.
For our sequence, we first identified the series as consisting of perfect squares: \(9, 16, 25, 36, 49\). These numbers correspond to the squares of consecutive integers: 3, 4, 5, 6, and 7.
Observing that each term increases as the integer it is based on increases by 1, we can write this pattern as:

• \(3^2 = 9\)
• \(4^2 = 16\)
• \(5^2 = 25\)
• \(6^2 = 36\)
• \(7^2 = 49\)

Each perfect square is built off the same base incrementally plus one: 3, 4, 5, etc. Recognizing sequence patterns like these leads directly into figuring out the general rule for finding any term in the sequence, also known as the general formula.

General Formula Derivation

Once we've identified the pattern that a sequence follows, the next step is to derive a general formula that can describe any term in the sequence.
Given our sequence 9, 16, 25, 36, 49, we determined each term is the square of a consecutive integer beginning at 3. To derive the general formula for a term \(a_n\):

• Start with the base integer, 3.
• Recognize that this base is incremented by its position in the sequence, \(n\), such that the first position (n=0) is \(3+0\).
• The formula becomes \((3 + n)^2\) because we are taking the square of a changing base integer.

Verifying this with each given term ensures the formula's accuracy. For \(n=0\), \(a_0 = (3+0)^2 = 9\); for \(n=1\), \(a_1 = (3+1)^2 = 16\), and so forth.
This formula provides a simple yet powerful tool to find any term in the sequence by merely substituting the term number for \(n\)."}]}]} Throughout history, the ability to recognize and generalize such patterns has been an essential part of mathematical discovery and problem-solving.