Problem 1 - Calculus for Biology and Medicine

Question

In Problems 1-16, determine the values of the sequence \(\left|a_{n}\right|\) for \(n=0,1,2, \ldots, 5\) $$ a_{n}=n+1 $$

Step-by-Step Solution

Answer

The sequence values are 1, 2, 3, 4, 5, 6.

Step 1: Define the sequence

The task is to determine the absolute values of the sequence \( |a_n| \) for \( n = 0, 1, 2, \ldots, 5 \). The sequence is given by \( a_n = n + 1 \).

Step 2: Calculate the sequence for each n

Substitute the values of \( n \) into the expression \( a_n = n + 1 \):- For \( n = 0 \), \( a_0 = 0 + 1 = 1 \).- For \( n = 1 \), \( a_1 = 1 + 1 = 2 \).- For \( n = 2 \), \( a_2 = 2 + 1 = 3 \).- For \( n = 3 \), \( a_3 = 3 + 1 = 4 \).- For \( n = 4 \), \( a_4 = 4 + 1 = 5 \).- For \( n = 5 \), \( a_5 = 5 + 1 = 6 \).

Step 3: Determine absolute values

The given sequence values are already positive, so the absolute values \( |a_n| \) remain the same:- \( |a_0| = 1 \)- \( |a_1| = 2 \)- \( |a_2| = 3 \)- \( |a_3| = 4 \)- \( |a_4| = 5 \)- \( |a_5| = 6 \)

Step 4: List the final results

The absolute values of the sequence \( |a_n| \) for \( n = 0, 1, 2, 3, 4, 5 \) are: 1, 2, 3, 4, 5, and 6.

Key Concepts

Absolute Value

Absolute value is a concept in mathematics that represents the distance of a number from zero, without considering the direction. This means whether the number is positive or negative, its absolute value is always a non-negative number. For example:

• The absolute value of 5 is 5, because it is 5 units away from zero on the number line.
• Similarly, the absolute value of -5 is also 5, because it's 5 units away from zero as well.

To denote absolute value, we use vertical bars around a number. If \(|a|\) represents the absolute value of \(a\), then \[|a| = a & \text{ if } a \geq 0 \|a| = -a & \text{ if } a < 0\]In our current problem, since the sequence given by \(a_n = n + 1\) naturally produces positive numbers for all specified terms (\(n = 0, 1, 2, \, \ldots, 5\)), the absolute values are the same as the original numbers.

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference of consecutive terms is constant. This difference is known as the "common difference". Recognizing an arithmetic sequence is straightforward:

• Start with the first term.
• Add the common difference to find the subsequent terms.

The general form of an arithmetic sequence can be expressed as:\[a_n = a_1 + (n - 1) imes d\]where \(a_1\) is the first term, \(d\) is the common difference, and \(n\) is the term number.
In the given problem, the sequence is given by \(a_n = n + 1\). This is indeed an arithmetic sequence where:

• The first term \(a_0 = 1\) (considering \(n = 0\) here to start)
• The common difference \(d = 1\)

This means each term increases by 1 as \(n\) increases by 1.

Calculating Sequence Terms

Calculating terms of a sequence involves substituting the value of \(n\) into the sequence's formula to determine the nth term. With practice, this process becomes intuitive and straightforward.
Given the sequence formula \(a_n = n + 1\), we find the sequence terms for specific values of \(n\) by direct substitution. For example:

• For \(n = 0\), substitute 0 for \(n\): \(a_0 = 0 + 1 = 1\)
• For \(n = 1\), substitute 1 for \(n\): \(a_1 = 1 + 1 = 2\)
• For \(n = 2\), substitute 2 for \(n\): \(a_2 = 2 + 1 = 3\)
• Continue this pattern for other values of \(n\)

This method can be applied to any arithmetic sequence to generate its terms. This process highlights how sequences can model ordered patterns or predict future terms from initial conditions.