Problem 1 - Thomas Calculus

Question

Each of Exercises \(1-6\) gives a formula for the \(n\) th term \(a_{n}\) of a sequence \(\left\\{a_{n}\right\\} .\) Find the values of \(a_{1}, a_{2}, a_{3},\) and \(a_{4} .\) $$ a_{n}=\frac{1-n}{n^{2}} $$

Step-by-Step Solution

Answer

The values are 0, -1/4, -2/9, -3/16.

Step 1: Calculate the first term

Substitute \( n = 1 \) into the formula \( a_n = \frac{1-n}{n^2} \).\[a_1 = \frac{1-1}{1^2} = \frac{0}{1} = 0\]The first term \( a_1 \) is 0.

Step 2: Calculate the second term

Substitute \( n = 2 \) into the formula \( a_n = \frac{1-n}{n^2} \).\[a_2 = \frac{1-2}{2^2} = \frac{-1}{4}\]The second term \( a_2 \) is \(-\frac{1}{4}\).

Step 3: Calculate the third term

Substitute \( n = 3 \) into the formula \( a_n = \frac{1-n}{n^2} \).\[a_3 = \frac{1-3}{3^2} = \frac{-2}{9}\]The third term \( a_3 \) is \(-\frac{2}{9}\).

Step 4: Calculate the fourth term

Substitute \( n = 4 \) into the formula \( a_n = \frac{1-n}{n^2} \).\[a_4 = \frac{1-4}{4^2} = \frac{-3}{16}\]The fourth term \( a_4 \) is \(-\frac{3}{16}\).

Key Concepts

nth term formula

Understanding the nth term formula is crucial when dealing with sequences. It essentially provides a way to express each term in a sequence as a function of its position number, "n." The formula allows us to calculate any term without needing to know the previous terms. This is particularly useful for sequences that have complex relationships between their terms.

In the exercise given, the nth term formula is expressed as:\[a_{n} = \frac{1-n}{n^2}\]Here, \(n\) represents the position of the term in the sequence (1st term, 2nd term, etc.).

The formula may look complex, but it simply combines basic arithmetic operations:

• The term \(1-n\) in the numerator involves subtraction, reducing the value as \(n\) increases.
• The denominator \(n^2\) simplifies the term by factoring the square of the position into the fraction.

Appreciating how these factors interact can make it easier to understand the sequence's structure and predict future terms.

calculation of sequence terms

Once we know the nth term formula, the next step is calculating individual terms in the sequence. This process involves substituting specific values for \(n\) into the formula.

Here's how you calculate the terms step-by-step:

• Term 1: Set \( n = 1 \), resulting in: \[a_1 = \frac{1-1}{1^2} = \frac{0}{1} = 0\]
• Term 2: Set \( n = 2 \), giving: \[a_2 = \frac{1-2}{2^2} = \frac{-1}{4}\]
• Term 3: Set \( n = 3 \), obtaining: \[a_3 = \frac{1-3}{3^2} = \frac{-2}{9}\]
• Term 4: Set \( n = 4 \), leading to: \[a_4 = \frac{1-4}{4^2} = \frac{-3}{16}\]

Notice how, as \(n\) increases, the terms generally become smaller in magnitude. This sequence gives us a glimpse into the behavior as each \(n\) changes. By substituting different values of \(n\), we can compute as many terms as needed.

evaluation of expressions

Evaluating expressions correctly is vital when working with sequence formulas. In our provided exercise, each term requires substituting a specific number for \(n\) and accurately performing the arithmetic.

Here are some key points to consider when evaluating:

• Maintain Order of Operations: Remember the BODMAS/BIDMAS principle to ensure calculations are correct: brackets, orders (powers and roots), division and multiplication, addition and subtraction.
• Manage Negative Signs: Be careful with subtraction and negative numbers, as they can easily lead to mistakes if not properly handled.
• Fraction Simplification: Work through the numerator and denominator separately before simplifying the fraction as needed.

Through careful attention to detail, we ensure the solution is accurate and valid, providing a precise value for each term in the sequence.