Problem 14 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 13-18: Find the specified term of the arithmetic sequence that has the two given terms. $$ a_{11} ; \quad a_{1}=2+\sqrt{2}, \quad a_{2}=3 $$

Step-by-Step Solution

Answer

The 11th term is \( 12 - 9\sqrt{2} \).

Step 1: Identify the arithmetic sequence formula

The general form of an arithmetic sequence is given by \( a_n = a_1 + (n-1) \cdot d \), where \( a_n \) is the \( n \)-th term, \( a_1 \) is the first term, and \( d \) is the common difference.

Step 2: Calculate the common difference

Use the given terms to find the common difference \( d \). We have \( a_1 = 2 + \sqrt{2} \) and \( a_2 = 3 \). Using the formula for the second term, \( 3 = 2 + \sqrt{2} + d \). Solving for \( d \), we find \( d = 3 - (2 + \sqrt{2}) = 1 - \sqrt{2} \).

Step 3: Use the formula to find \( a_{11} \)

Substitute the values of \( a_1 \) and \( d \) back into the arithmetic sequence formula to find \( a_{11} \). \( a_{11} = (2 + \sqrt{2}) + (11-1)(1-\sqrt{2}) \).

Step 4: Simplify the expression

Simplify the expression: \[ a_{11} = (2 + \sqrt{2}) + 10 \cdot (1-\sqrt{2}) = 2 + \sqrt{2} + 10 - 10\sqrt{2} \]. Combine like terms: \[ = 12 - 9\sqrt{2} \].

Key Concepts

Common Difference

In arithmetic sequences, the common difference is a fundamental concept. It represents the consistent increase or decrease between consecutive terms in a sequence. In simpler terms, it's what you add or subtract every time to get to the next number. Finding this difference is crucial because it allows you to predict further terms easily.

To calculate the common difference (\( d \)), utilize known terms of the sequence. In the exercise, the first term (\( a_1 \)) is given as \( 2 + \sqrt{2} \), and the second term (\( a_2 \)) is \( 3 \). The formula to find \( d \) can be applied as:

• \( a_2 = a_1 + d \)
• Substitute the values: \( 3 = 2 + \sqrt{2} + d \)
• Solving it gives: \( d = 1 - \sqrt{2} \)

This crucial value of \( d \) assists in calculating other terms in the sequence efficiently.

Sequence Formula

The sequence formula is a valuable formula for arithmetic sequences. Represented as \( a_n = a_1 + (n-1) \cdot d \), it helps find the value of any term (\( a_n \)) in the sequence, knowing the first term (\( a_1 \)) and the common difference (\( d \)).

This formula breaks down as:

• \( a_n \): Desired term you wish to find.
• \( a_1 \): The initial term of the sequence.
• \( (n-1) \cdot d \): Amount you need to add to the first term to reach the nth term.

Using this formula is straightforward once you have \( a_1 \) and \( d \). For example, to find the 11th term (\( a_{11} \)) in our sequence, substitute the values into the formula:

• \( a_{11} = (2 + \sqrt{2}) + (11-1)(1-\sqrt{2}) \)
• The calculation simplifies further to help us find that specific term.

Term Calculation

Calculating a specific term in an arithmetic sequence involves substituting values into the sequence formula. To find \( a_{11} \), you'll follow some simple steps:

1. Identify the formula: \( a_n = a_1 + (n-1)d \).2. Use known values: given \( a_1 = 2 + \sqrt{2} \) and \( d = 1 - \sqrt{2} \), determine \( a_{11} \).
Substituting:

• \( a_{11} = (2 + \sqrt{2}) + 10(1-\sqrt{2}) \)
• Simplify: \( 2 + \sqrt{2} + 10 - 10\sqrt{2} \)
• Combine like terms to find \( a_{11} = 12 - 9\sqrt{2} \)

Understanding each element of this process allows for smooth calculation of other terms, illustrating the usefulness of the sequence formula and understanding the common difference.