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Chapter 6: Problem 35

Find the \(n\) th term of the arithmetic sequence. $$-8,-5,-2, \ldots$$

Short Answer

Expert verified
The \(n\)th term of the arithmetic sequence is \(3n - 11\)

Step by step solution

01

Identify the first term and the common difference

To identify the first term, simply take the first number from the sequence, which is -8. Then find the common difference in the sequence. It is the difference between any two successive terms. Here, the common difference \(d = -5 - (-8) = 3\). So the first term \(a_1 = -8\) and the common difference \(d = 3\)
02

Plug the values into the formula and simplify

Now that we have the first term and the common difference, we can use the formula for the \(n\) th term of an arithmetic sequence which is \(a_n = a_1 + (n-1) \cdot d\). By substituting \(a_1 = -8\) and \(d = 3\) into the equation, we get \(a_n = -8 + (n-1) \cdot 3\). Expanding this, we get \(a_n = -8 + 3n - 3 = 3n - 11\). Thus, the \(n\) th term of the sequence is \(3n - 11\)

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Key Concepts

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

nth Term Formula
The nth term formula for an arithmetic sequence is a key tool that allows us to find any term in the sequence without listing all the terms. This formula is expressed as \( a_n = a_1 + (n-1) \cdot d \), where:
  • \( a_n \) is the nth term that you are trying to find.
  • \( a_1 \) is the first term of the arithmetic sequence.
  • \( n \) is the term number.
  • \( d \) is the common difference between consecutive terms.
Applying this formula is simple once you have the first term and the common difference. Plug these values into the formula to calculate any term number. This saves time and maintains accuracy, especially for large values of \( n \). Understanding this fundamental tool can greatly aid in working with sequences more efficiently.
Common Difference
The common difference \( d \) in an arithmetic sequence is the amount you add (or subtract if it's negative) to get from one term to the next. It represents the consistent change between consecutive terms throughout the sequence.
  • To find \( d \), simply subtract the first term from the second term \( d = a_2 - a_1 \).
  • The sequence can be increasing, decreasing, or constant depending on whether \( d \) is positive, negative, or zero.
In our example, the sequence is \(-8, -5, -2, \ldots\), and the common difference is 3. We calculated it by \(-5 - (-8) = 3\). This consistent difference not only helps in identifying the pattern of the sequence but also in utilizing the nth term formula effectively.
Sequence Formulation
Sequence formulation involves establishing a clear mathematical expression that represents all elements of the sequence. For arithmetic sequences, it focuses on building a formula which uses the first term and the common difference. This enables the calculation of any term within the sequence.
  • Begin by identifying the first term \( a_1 \), which is the starting point of your sequence.
  • Determine \( d \), the common difference, to understand how the sequence progresses.
  • Use these values in the nth term formula \( a_n = a_1 + (n-1) \cdot d \) to create the expression for the sequence.
In our exercise, the formulation yielded \( a_n = 3n - 11 \), enabling easy computation of any desired term number. This structured approach clarifies how the sequence develops and provides a straightforward method to find subsequent terms without manual calculation.

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Most popular questions from this chapter

Find the \(n\)th partial sum of the arithmetic sequence. $$a_{n}=n-4 ; n=25$$

Find the sum of the geometric series. $$\sum_{n=1}^{14}\left(\frac{4}{3}\right)^{n}$$

Find the \(n\) th term of the geometric sequence. $$0.5,0.05,0.005, \ldots$$

If \(f(x)\) is a linear polynomial, show that \(f(n),\) where \(n\) is a positive integer, is an arithmetic sequence.

Find the sum of the infinite geometric series. $$\sum_{n=1}^{\infty}\left(-\frac{3}{5}\right)^{n}$$
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