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Chapter 19: Problem 10

Find the nth term of the arithmetic sequence with the given values. $$2,0.5,-1, \ldots ; n=25$$

Short Answer

Expert verified
The 25th term of the sequence is -34.

Step by step solution

01

Identify the Common Difference

The common difference, usually denoted as \( d \), is the difference between successive terms in the sequence. We can find it by subtracting the first term from the second term: \( 0.5 - 2 = -1.5 \). For verification, subtract the second term from the third term: \( -1 - 0.5 = -1.5 \). Thus, the common difference \( d \) is \( -1.5 \).
02

Use the Formula for the nth Term

The formula for the nth term \( a_n \) of an arithmetic sequence is given by: \( a_n = a_1 + (n-1) imes d \), where \( a_1 \) is the first term, \( n \) is the number of the term, and \( d \) is the common difference. Here, \( a_1 = 2 \), \( d = -1.5 \), and \( n = 25 \).
03

Substitute Known Values into the Formula

Substitute the known values into the formula: \[ a_{25} = 2 + (25 - 1) imes (-1.5) \]. This simplifies to \[ a_{25} = 2 + 24 imes (-1.5) \].
04

Calculate the 25th Term

Calculate \( 24 imes (-1.5) \); this equals \( -36 \). Then, substitute back to find \( a_{25} \): \[ a_{25} = 2 - 36 = -34 \]. Thus, the 25th term of the sequence is \( -34 \).

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Key Concepts

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

Understanding Common Difference
In an arithmetic sequence, the **common difference** is a crucial element that determines the increase or decrease between consecutive terms. When you look at consecutive terms in a sequence, the common difference is found consistently across the sequence. To find the common difference, subtract any term from its succeeding term. For example:
  • In the sequence 2, 0.5, -1, ..., find the difference:
  • Subtract the first term (2) from the second term (0.5): \(0.5 - 2 = -1.5\).
To confirm your calculation, check another pair of consecutive terms:
  • Subtract the second term (0.5) from the third term (-1): \(-1 - 0.5 = -1.5\).
This confirms that the common difference \(d\) is indeed \(-1.5\). Every term in this arithmetic sequence decreases by 1.5, clearly reflecting the defined common difference.
Exploring the nth Term Formula
The **nth term formula** allows you to find any term in an arithmetic sequence without listing all previous terms. This formula is valuable because it provides a direct path to finding any term in a sequence once you have certain foundational numbers. The formula to find the nth term \(a_n\) is given by:
  • \[a_n = a_1 + (n-1) \times d\]
  • where \(a_1\) is the first term of the sequence, \(n\) is the term number, and \(d\) is the common difference.
For example, if the first term \(a_1\) is 2, and the common difference \(d\) is \(-1.5\), to find the 25th term:
  • Substitute the values into the formula: \[a_{25} = 2 + (25 - 1) \times (-1.5)\]
  • This simplifies to: \[a_{25} = 2 + 24 \times (-1.5)\]
This approach saves time and avoids the potential for error that could occur if you had to calculate each term up to the 25th one manually.
Understanding Mathematical Sequences
**Mathematical sequences** are ordered sets of numbers that follow a specific pattern. An arithmetic sequence is a kind of sequence characterized by a constant difference鈥攌nown as the common difference鈥攂etween consecutive terms. These sequences can increase or decrease based on the sign of this common difference.
For arithmetic sequences, key characteristics to remember include:
  • The sequence is usually represented as \(a_1, a_2, a_3, \ldots\), where each subsequent term is formed by adding the common difference \(d\) to the prior term.
  • In our example sequence 2, 0.5, -1, ..., each subsequent value is decreased by 1.5, which is the calculated common difference.
  • The sequence continues infinitely, but formulas like the nth term formula help pinpoint specific terms efficiently.
Understanding these basics about mathematical sequences and the how-to behind arithmetic sequences sets a strong foundation for exploring more advanced topics in mathematics.

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Most popular questions from this chapter

Find the fractions equal to the given decimals. $$0.499999 \ldots$$

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