Problem 36 - Algebra and Trigonometry

Question

\(27-36\) . Determine the common difference, the fifth term, the nth term, and the 100 th term of the arithmetic sequence. $$ -t,-t+3,-t+6,-t+9, \ldots $$

Step-by-Step Solution

Answer

The common difference is 3, the fifth term is \(-t + 12\), the nth term is \(-t + 3n - 3\), and the 100th term is \(-t + 297\).

Step 1: Determine the common difference

In an arithmetic sequence, the common difference is defined as the difference between successive terms. Let's calculate the common difference by subtracting the first term from the second term: \(-t+3 - (-t) = -t+3 + t = 3\).Thus, the common difference is 3.

Step 2: Find the fifth term of the sequence

To find the fifth term of an arithmetic sequence, we can use the formula for the nth term: \(a_n = a_1 + (n-1) imes d\), where \(a_1\) is the first term, \(n\) is the term number, and \(d\) is the common difference.For the fifth term: \(a_5 = -t + (5-1) \times 3\)\(a_5 = -t + 12\).The fifth term of the sequence is \(-t + 12\).

Step 3: Derive the formula for the nth term

The nth term of an arithmetic sequence is given by:\(a_n = a_1 + (n-1) imes d\).Since \(a_1 = -t\) and the common difference \(d = 3\):\(a_n = -t + (n-1) imes 3\).Simplifying this gives:\(a_n = -t + 3n - 3\).Thus, the formula for the nth term is: \(a_n = -t + 3n - 3\).

Step 4: Calculate the 100th term

Use the nth term formula derived in the previous step to find the 100th term:\(a_{100} = -t + (3 \times 100) - 3\)\(a_{100} = -t + 300 - 3\)\(a_{100} = -t + 297\).The 100th term is \(-t + 297\).

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is a crucial component. It is the constant amount that each term increases (or decreases) by as you move from one term to the next. Understanding this helps to generate all the terms of the sequence.

To find the common difference, you subtract the first term from the second term. For the sequence given in the exercise, the first term is \(-t\) and the second term is \(-t+3\). By subtracting these, \(-t+3 - (-t) = 3\), we find that the common difference is 3.

Having a positive common difference of 3 means that each term in the sequence is 3 units greater than the term before it. This concept helps in predicting subsequent terms and in understanding the sequence pattern.

Nth Term Formula

The nth term formula in an arithmetic sequence is used to find any term in the sequence without listing all the previous terms. It creates a shortcut to directly find the value of any term in the sequence.

The general formula is given by: \(a_n = a_1 + (n-1) \times d\). Here, \(a_1\) is the first term, \(n\) is the term number, and \(d\) is the common difference. Applying this formula helps in calculating any term in the sequence efficiently.

In our sequence: \(a_1 = -t\) and \(d = 3\). Plugging these values into the formula, we simplify the expression to \(a_n = -t + 3n - 3\). This formula simplifies finding any term in the sequence and shows the dependency of each term on its position, \(n\), and our constant \(-t\).

Fifth Term

Finding the fifth term is straightforward once we have the nth term formula. We plug \(n = 5\) into our simplified formula to determine the fifth term of the sequence.

Using the formula: \(a_5 = -t + 3(5) - 3\).
Breaking down the calculation:

• Multiply: \(3 \times 5 = 15\).
• Subtract: \(15 - 3 = 12\).
• Add: \(-t + 12\).

Consequently, the fifth term is \(-t + 12\). This shows how the arithmetic sequence grows by fixed steps, allowing us to predict and calculate terms systematically.

100th Term

The 100th term can be calculated using the nth term formula just like any other term. It exemplifies how powerful and useful this formula is, especially when dealing with large sequences.

To find it, set \(n = 100\) in the formula \(a_n = -t + 3n - 3\).
Calculation steps:

• Multiply: \(3 \times 100 = 300\).
• Subtract: \(300 - 3 = 297\).
• Add: \(-t + 297\).

Thus, the 100th term of the sequence is \(-t + 297\). With each consistent step—each common difference of 3—the sequence allows us to calculate incredibly large terms with ease.