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Chapter 6: Problem 39

Find the \(n\) th term of the arithmetic sequence. $$a, a+2, a+4, \ldots$$

Short Answer

Expert verified
The nth term of the arithmetic sequence is \(a_n = a + 2n - 2\).

Step by step solution

01

Identify the first term and common difference

In the given sequence \(a, a+2, a+4, \ldots\), the first term \(a_1\) is 'a' and the common difference \(d\) is 2.
02

Apply the formula for the nth term of an arithmetic sequence

The nth term \(a_n\) of an arithmetic sequence can be found using the formula \(a_n = a_1 + (n - 1) * d\). Substituting the given values, we get \(a_n = a + (n - 1) * 2\).
03

Simplify the equation

Simplify the equation to find the nth term. So, \(a_n = a + 2n - 2\). Hence, the nth term of the given arithmetic sequence is \(a_n = a + 2n - 2\).

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Key Concepts

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

nth term formula
When dealing with arithmetic sequences, the nth term formula is a key tool. This formula allows us to find any term's value in the sequence without having to list all preceding terms. For an arithmetic sequence, the nth term formula is given by:
  • \( a_n = a_1 + (n - 1) \cdot d \)
Here:
  • \( a_n \) represents the term we want to find.'
  • \( a_1 \) is the first term of the sequence.
  • \( d \) is the common difference between successive terms.
  • \( n \) is the position of the term in the sequence that we want to calculate.
This formula is a simple yet powerful way to explore arithmetic sequences. It stems from understanding that each term in the sequence builds upon the previous one by adding the common difference. By applying this formula, any term in the sequence can be easily calculated.
common difference
The common difference is a fundamental part of an arithmetic sequence. It is defined as the fixed amount that each term in the sequence increases (or decreases) by.
  • To find the common difference, simply subtract any term in the sequence from the term that follows it.
For example, in the sequence \( a, a+2, a+4, \ldots \), the common difference is:
  • \( a+2 - a = 2 \)
This tells us each term is 2 more than the previous term. The consistency of the common difference is what makes the sequence arithmetic.
  • The common difference provides a simple mechanism to predict future terms.
  • If the common difference is positive, the sequence grows.
  • If the common difference is negative, the sequence decreases.
first term
Identifying the first term in an arithmetic sequence is crucial, as it forms the foundation from which all subsequent terms are calculated. The first term is often denoted as \( a_1 \).
  • In the sequence \( a, a+2, a+4, \dots \), the first term \( a_1 \) is simply \( a \).
In arithmetic sequences, knowing the first term is vital because it's the initial stepping stone for building up the sequence with the common difference. Once you know the first term and the common difference, you can use the nth term formula to find any other term in the sequence.
  • Understanding this concept ensures you correctly apply the formula for the nth term.
  • The first term often sets the pattern and rhythm of your sequence.
Both the first term and the common difference give the sequence its unique character and help in making precise calculations of any term within it.

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