Problem 7 - Calculus for Biology and Medicine

Question

Determine the values of the sequence \(\left\\{a_{n}\right\\}\) for \(n=0,1,2, \ldots, 5\) $$ f(n)=(n+1)^{2} $$

Step-by-Step Solution

Answer

The sequence values are 1, 4, 9, 16, 25, 36.

Step 1: Understand the Given Function

The given function is \( f(n) = (n+1)^2 \). This means for each integer value of \( n \), we will compute \( (n+1)^2 \).

Step 2: Compute f(0)

Substitute \( n = 0 \) into the function: \[ f(0) = (0+1)^2 = 1^2 = 1 \] Thus, \( a_0 = 1 \).

Step 3: Compute f(1)

Substitute \( n = 1 \) into the function: \[ f(1) = (1+1)^2 = 2^2 = 4 \] Thus, \( a_1 = 4 \).

Step 4: Compute f(2)

Substitute \( n = 2 \) into the function: \[ f(2) = (2+1)^2 = 3^2 = 9 \] Thus, \( a_2 = 9 \).

Step 5: Compute f(3)

Substitute \( n = 3 \) into the function: \[ f(3) = (3+1)^2 = 4^2 = 16 \] Thus, \( a_3 = 16 \).

Step 6: Compute f(4)

Substitute \( n = 4 \) into the function: \[ f(4) = (4+1)^2 = 5^2 = 25 \] Thus, \( a_4 = 25 \).

Step 7: Compute f(5)

Substitute \( n = 5 \) into the function: \[ f(5) = (5+1)^2 = 6^2 = 36 \] Thus, \( a_5 = 36 \).

Key Concepts

Function Evaluation

Function evaluation involves calculating the output of a given function for specific input values. In this exercise, the function is \( f(n) = (n+1)^2 \). This mathematical expression tells us that for every integer \( n \), we simply add 1 to \( n \) and then square the result. Evaluating a function means substituting different values of \( n \) into the function to find corresponding outputs.

• For \( n = 0 \), the function is \( f(0) = (0+1)^2 = 1^2 = 1 \).
• When \( n = 1 \), it becomes \( f(1) = (1+1)^2 = 2^2 = 4 \).
• Following same steps, you perform similar calculations for \( n = 2, 3, 4,\) and \(5 \).

The functional evaluation step is crucial as it computes distinct numeric results called the sequence's terms.

Integer Sequence

An integer sequence is a list of numbers where each number is directly related to its position in the list, often determined by applying rules or functions. In this problem, the rule is \( f(n) = (n+1)^2 \), creating a sequence by evaluating this function for integers from 0 to 5.The sequence derived here is:

• \( a_0 = 1 \)
• \( a_1 = 4 \)
• \( a_2 = 9 \)
• \( a_3 = 16 \)
• \( a_4 = 25 \)
• \( a_5 = 36 \)

Splitting these values into a clear list helps identify patterns, such as recognizing the sequence as the square numbers starting from 1 squared. Patterns in integer sequences assist in predicting future values and understanding the sequence's nature.

Step-by-Step Solution

Breaking down problems into smaller, manageable steps is vital in mathematics, enhancing clarity and solving complex problems efficiently. The step-by-step approach follows a logical sequence of actions, making the solution more accessible.Here's how the exercise was tackled step-by-step:

• Firstly, the function's rule \( f(n) = (n+1)^2 \) is comprehended. Each computation takes the place of the corresponding integer \( n \).
• Substitute \( n = 0 \) through \( n = 5 \) individually into the function to generate output values, hence creating the sequence.
• This involves a clear understanding of algebraic manipulation, such as addition and squaring numbers, to ensure accuracy in each step.

By putting emphasis on each stage, from understanding the rule to calculating each result, step-by-step solutions demystify the changes and outputs, making problem-solving more approachable for learners.