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Chapter 9: Problem 14

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

Short Answer

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

Step by step solution

01

Understand the Arithmetic Sequence Formula

The general formula for finding the nth term of an arithmetic sequence is \( a_n = a_1 + (n-1) \cdot d \), where \( a_1 \) is the first term and \( d \) is the common difference.
02

Find the Common Difference

Use the first two terms to find the common difference \( d \). \[ d = a_2 - a_1 = 3 - (2 + \sqrt{2}) = 1 - \sqrt{2} \].
03

Use the Common Difference to Find the Specified Term

Given that \( a_{11} \) is needed, plug in the values: \[ a_{11} = 2 + \sqrt{2} + (11-1) \cdot (1 - \sqrt{2}) \]Calculate it step-by-step: \( a_{11} = 2 + \sqrt{2} + 10 \cdot (1 - \sqrt{2}) \).Simplifying further gives: \( a_{11} = 2 + \sqrt{2} + 10 - 10\sqrt{2} \).
04

Simplify the Expression

Combine like terms: \( a_{11} = 12 - 9\sqrt{2} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Difference
In an arithmetic sequence, a key idea to grasp is the 'common difference'. This is the consistent gap between consecutive terms.
Let's say you have a sequence starting with the first term, known as \( a_1 \), followed by the second term \( a_2 \). To find the common difference \( d \), you simply subtract the first term from the second:
  • \( d = a_2 - a_1 \)
For example, if \( a_1 = 2 + \sqrt{2} \) and \( a_2 = 3 \), the common difference is found by:
  • \( d = 3 - (2 + \sqrt{2}) = 1 - \sqrt{2} \)
Spotting this consistent difference across all terms helps predict future terms or calculate terms further out in the sequence.
nth Term Formula
When exploring arithmetic sequences, one formula you will frequently use is the nth term formula. This formula gives you any term's value within a sequence without needing to list all prior terms.
This formula is expressed as:
  • \( a_n = a_1 + (n - 1) \cdot d \)
With this, \( a_1 \) is your initial term, \( n \) is the term's position in the sequence, and \( d \) is the common difference you would have previously calculated.
By using this structure, you can directly plug in values to find specific terms, such as how to find \( a_{11} \) using \( a_1 = 2 + \sqrt{2} \) and \( d = 1 - \sqrt{2} \).
Sequence Term Calculation
Calculating a specific term within an arithmetic sequence relies on the nth term formula. In our specific example, we're looking to discover the 11th term, \( a_{11} \).
Using the nth term formula:
  • \( a_{11} = a_1 + (11 - 1) \cdot d \)
You substitute the known values, \( a_1 = 2 + \sqrt{2} \) and \( d = 1 - \sqrt{2} \), into this equation:
  • \( a_{11} = 2 + \sqrt{2} + 10 \cdot (1 - \sqrt{2}) \)
This setup allows for calculating \( a_{11} \) directly without having to calculate each intermediate term.
Simplifying Expressions
Once you've used the formula to find a sequence term, the next step is simplifying the expression. This helps make the result more understandable.
In the example we've been reviewing:
  • \( a_{11} = 2 + \sqrt{2} + 10 - 10\sqrt{2} \)
You'll combine like terms. You have constants (2 and 10), and like radical terms (\( \sqrt{2} \) and \( -10\sqrt{2} \)). Combine these separately:
  • Constants: \( 2 + 10 = 12 \)
  • Radicals: \( \sqrt{2} - 10\sqrt{2} = -9\sqrt{2} \)
Thus, simplifying gives you:
  • \( a_{11} = 12 - 9\sqrt{2} \)
This process ensures clarity in the final response.

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