Problem 27 - Calculus for Biology and Medicine

Question

Find an expression for \(a_{n}\) on the basis of the values of \(a_{0}, a_{1}, a_{2}, \ldots .\) $$ 1,2,4,8,16, \ldots $$

Step-by-Step Solution

Answer

The expression for \(a_{n}\) is \(a_{n} = 2^n\).

Step 1: Identify the Pattern

Observe the given sequence: 1, 2, 4, 8, 16, ... Each term appears to be a power of 2. Specifically, these are 2^0, 2^1, 2^2, 2^3, 2^4, respectively.

Step 2: Establish the General Formula

Based on the pattern, the n-th term of the sequence can be expressed as a power of 2. Thus, the general formula for the n-th term is \( a_{n} = 2^n \).

Step 3: Verify the Formula

Check the formula against a few terms in the sequence: \( a_{0} = 2^0 = 1 \), \( a_{1} = 2^1 = 2 \), \( a_{2} = 2^2 = 4 \), \( a_{3} = 2^3 = 8 \), and \( a_{4} = 2^4 = 16 \). The formula matches the sequence.

Key Concepts

Exponential Growth

Exponential growth describes a process where the rate of growth is proportional to the current quantity, resulting in quantities that grow faster and faster over time. In the given sequence, each term is twice the previous one, which is a hallmark of exponential growth.

This means that instead of adding a fixed number to reach the next term, we multiply by a constant factor—here that factor is 2. This results in a rapid increase, or growth, of the sequence. This kind of growth can be commonly observed in many natural and financial processes, such as population growth or compound interest.

Understanding exponential growth is crucial because it helps us predict how things will grow over time when they expand at a constant rate. In math, we often represent this kind of growth with formulas involving powers or exponents, like our example which uses powers of 2.

Pattern Recognition

Pattern recognition in sequences involves observing and identifying regularities or trends. By examining a series of numbers, one can determine a pattern that describes the relationship between the numbers. In this sequence, we see numbers like 1, 2, 4, 8, 16, which can initially appear random until we notice that each term is a power of 2.

Recognizing this pattern helps us simplify the problem and come up with a general formula. By identifying the core pattern, we can make predictions about future terms in the sequence with confidence. Patterns help us see underlying rules and structures, which are critical for solving mathematical problems efficiently.

When faced with sequences, always look for multiplication or addition patterns, or how terms relate to one another. This often provides the key insight needed for developing a solution.

Recursive Formula

A recursive formula expresses each term in a sequence as a function of its previous terms. This approach is useful for sequences where there’s a clear, repetitive process to generate subsequent numbers based on earlier ones.

For the sequence 1, 2, 4, 8, 16, a recursive formula can be written as:

• If the first term is given: \(a_0 = 1\)
• The recursive formula would be: \(a_{n} = 2 \times a_{n-1}\)

Given \(a_0 = 1\), we can use this formula to calculate any term, as it simply requires multiplying the previous term by 2.

This recursive relationship reflects the constant multiplication pattern that drives exponential growth. Being able to describe sequences recursively can be very helpful, especially when programming, as it represents a direct step-by-step process for generating sequence elements.