Problem 6 - 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 \(a_{4} .\) $$ a_{n}=\frac{2^{n}-1}{2^{n}} $$

Step-by-Step Solution

Answer

The terms are: \(a_1 = \frac{1}{2}\), \(a_2 = \frac{3}{4}\), \(a_3 = \frac{7}{8}\), \(a_4 = \frac{15}{16}\).

Step 1: Find the first term

To find the first term, substitute \(n = 1\) into the formula: \(a_1 = \frac{2^1 - 1}{2^1} = \frac{2 - 1}{2} = \frac{1}{2}\). Thus, the value of \(a_1\) is \(\frac{1}{2}\).

Step 2: Find the second term

For the second term, substitute \(n = 2\) into the formula: \(a_2 = \frac{2^2 - 1}{2^2} = \frac{4 - 1}{4} = \frac{3}{4}\). The value of \(a_2\) is \(\frac{3}{4}\).

Step 3: Find the third term

Calculate the third term by inserting \(n = 3\): \(a_3 = \frac{2^3 - 1}{2^3} = \frac{8 - 1}{8} = \frac{7}{8}\). The value of \(a_3\) is \(\frac{7}{8}\).

Step 4: Find the fourth term

Substitute \(n = 4\) into the sequence formula: \(a_4 = \frac{2^4 - 1}{2^4} = \frac{16 - 1}{16} = \frac{15}{16}\). Therefore, \(a_4\) is \(\frac{15}{16}\).

Key Concepts

nth term

A sequence is a list of numbers arranged in a specific order, and each number in that sequence is called a term. To find any term in a sequence, we often use a formula that defines the nth term. The nth term formula allows us to find any term without having to list out all the previous terms.

In our exercise, the formula is given as \(a_{n} = \frac{2^n - 1}{2^n}\). The n represents the position of the term in the sequence. It’s like having a key that unlocks the secret to finding the right number in the sequence. For instance, if you were asked to find the 100th term, you could plug 100 into the formula instead of listing terms 1 to 99. This makes calculations not only fast but incredibly efficient.

Understanding the nth term is crucial because it provides a mathematical way to analyze trends and patterns in data, which is a skill applicable in many areas, such as computing and data analysis.

sequence calculation

Sequence calculation involves plugging the appropriate numbers into the sequence's nth term formula to compute the value of specific terms. It's like following a recipe: step-by-step, you calculate each term.

For this specific exercise, we performed several sequence calculations by substituting different n values into the formula \(a_{n} = \frac{2^n - 1}{2^n}\):

• By substituting \(n = 1\), we calculated the first term \(a_1 = \frac{1}{2}\).
• For \(n = 2\), the second term became \(a_2 = \frac{3}{4}\).
• With \(n = 3\), the third term was calculated to be \(a_3 = \frac{7}{8}\).
• Lastly, substituting \(n = 4\) gave \(a_4 = \frac{15}{16}\).

By using the formula, these calculations ensure accuracy and provide insights into how terms develop as the sequence progresses. The sequence calculation process can also unveil relationships between successive terms, such as increasing patterns or ratios.

term computation

Term computation refers to the specific process of calculating exact values for terms in a sequence. It is a methodical approach that ensures precise numerical results.

In this exercise, term computation was performed by following a systematic approach: substitution, simplification, and solving the formula. For instance:

• First, you substitute the desired n value into the nth term formula.
• Next, you simplify the powers of 2 and perform arithmetic operations like subtraction and division.
• Finally, you arrive at the computed term by solving the simplified expression.

A term computation not only zeroes in on the exact value but also strengthens your mathematical fluency by providing a deeper understanding of how expressions and numbers interact in sequences. Moreover, this skill is essential when dealing with small and large-scale numerical problems across various fields of study.