Problem 23 - Algebra and Trigonometry

Question

21-26 \(\approx\) Find the first five terms of the sequence, and determine whether it is arithmetic. If it is arithmetic, find the common difference, and express the \(n\) th term of the sequence in the standard form \(a_{n}=a+(n-1) d\) $$ a_{n}=\frac{1}{1+2 n} $$

Step-by-Step Solution

Answer

The sequence is not arithmetic; differences between terms vary.

Step 1: Substitute Values for the First Five Terms

To find the first five terms of the sequence, substitute values of \( n \) from 1 to 5 into the sequence formula \( a_n = \frac{1}{1+2n} \):\[a_1 = \frac{1}{1+2(1)} = \frac{1}{3}, \a_2 = \frac{1}{1+2(2)} = \frac{1}{5}, \a_3 = \frac{1}{1+2(3)} = \frac{1}{7}, \a_4 = \frac{1}{1+2(4)} = \frac{1}{9}, \a_5 = \frac{1}{1+2(5)} = \frac{1}{11}\]

Step 2: Determine If the Sequence is Arithmetic

An arithmetic sequence has a constant difference between consecutive terms. Calculate the difference between each consecutive pair of terms:\[a_2 - a_1 = \frac{1}{5} - \frac{1}{3} = -\frac{2}{15}, \a_3 - a_2 = \frac{1}{7} - \frac{1}{5} = -\frac{2}{35}, \a_4 - a_3 = \frac{1}{9} - \frac{1}{7} = -\frac{2}{63}, \a_5 - a_4 = \frac{1}{11} - \frac{1}{9} = -\frac{2}{99}\]Since the differences are not constant, the sequence is not arithmetic.

Step 3: Conclusion

Since the sequence is not arithmetic, there is no common difference and the formula for the nth term \( a_{n} = a + (n-1) d \) is not applicable in this case. The sequence is determined entirely by \( a_n = \frac{1}{1+2n} \).

Key Concepts

Arithmetic Sequences

Arithmetic sequences are a fundamental concept in mathematics, characterized by a constant difference between successive terms. To identify an arithmetic sequence, simply subtract the preceding term from the following term. If this difference remains consistent throughout the sequence, it is arithmetic.

These sequences can be easily recognized by the way numbers seem to add up at the same rate. Arithmetic sequences are handy in everyday calculations, such as predicting patterns or getting the total of evenly spaced numbers. Whether you're tallying up costs or just lining up a row of numbers, arithmetic sequences help simplify things.

Common Difference

The common difference in an arithmetic sequence is the fixed amount you add (or subtract) to get from one term to the next. It provides a simple way of building a sequence out of thin air, once you have a starting point, or the first term.

To find the common difference, subtract any term from the one after it. For example, if the second term is 7 and the first term is 4, the common difference is 3 because, \(7 - 4 = 3\). Knowing the common difference is key to predicting what comes next in the sequence and understanding its trend.

Having a constant step makes arithmetic sequences predictable, and that's what makes them a vital tool in math and other fields like physics and engineering. It's like following a trail where each step is the same length.

Nth Term Formula

The nth term formula is your mathematical map to find any term in an arithmetic sequence without listing all previous terms. This is especially beneficial when dealing with sequences that have many terms.

The formula is expressed as \( a_n = a + (n-1)d \), where:

• \(a\) represents the first term,
• \(d\) is the common difference,
• and \(n\) is the term number you wish to find.

Using this formula, you can skip right to any term number you want, saving time and effort. For instance, finding the 50th or 100th term becomes just a quick calculation, rather than a long sequence of additions. Having this formula makes it easier to handle tasks involving arithmetic sequences efficiently.

Sequence Analysis

Sequence analysis is about exploring the storyline of numbers in a sequence. In the context of arithmetic sequences, this involves checking properties like regularity and pattern repetition. The initial step is to determine whether the sequence is arithmetic by checking if there’s a constant common difference.

Take, for example, the sequence \(\frac{1}{3}, \frac{1}{5}, \frac{1}{7}, \frac{1}{9}, \frac{1}{11}\). By analyzing the differences between these terms, you notice they're not the same, which tells us this sequence is not arithmetic. This kind of analysis is crucial for determining what type of sequence you are dealing with and deciding on the proper methods to work with it.

In broader terms, sequence analysis helps in understanding mathematical trends, forecasting future values, and forming connections between different mathematical ideas. It's about taking a closer look at a sequence to make sense of its unique properties and use those insights effectively.