Problem 17
Question
Write the next two apparent terms of the sequence. Describe the pattern you used to find these terms. \(5,10,20,40, \ldots\)
Step-by-Step Solution
Verified Answer
The next two terms of the sequence are 80 and 160. The pattern used to find these terms is that each term of the sequence is double the preceding term.
1Step 1: Identifying the Pattern
Observe the sequence and identify the pattern. Every term of the sequence is double the previously mentioned term. This means the sequence follows a geometric progression where the common ratio is 2.
2Step 2: Applying the Pattern to find the next term
Using this pattern, multiply the last known term in the sequence (which is 40) by 2 to find the next term. That is, \(40 \times 2 = 80\)
3Step 3: Continue Pattern to Find the Subsequent term
To find the subsequent term, multiply 80 by 2. That is, \(80 \times 2 = 160\)
Key Concepts
Arithmetic SequencesMathematical PatternsSequence Analysis
Arithmetic Sequences
In mathematics, an arithmetic sequence is a sequence of numbers where each term after the first is found by adding a constant to the previous term. This constant is known as the common difference. For example, in the sequence 3, 6, 9, 12, each term increases by 3, which is the common difference.
To identify an arithmetic sequence, look for a consistent difference between consecutive terms. This characteristic makes arithmetic sequences predictable and easy to extend by simply adding the common difference to the latest term.
To identify an arithmetic sequence, look for a consistent difference between consecutive terms. This characteristic makes arithmetic sequences predictable and easy to extend by simply adding the common difference to the latest term.
- If a sequence is not arithmetic, reassess the difference or explore other sequence types like geometric sequences.
- Arithmetic sequences are linear patterns, graphing them will show a straight line.
Mathematical Patterns
Identifying mathematical patterns in sequences is essential for understanding and predicting future terms. Patterns can be arithmetic, geometric, or even more complex, making it crucial to recognize the type of pattern present.
In the exercise sequence 5, 10, 20, 40, the pattern is geometric, as each term is double the previous term. Recognizing this pattern makes it easy to predict future terms by consistently applying the identified rule.
In the exercise sequence 5, 10, 20, 40, the pattern is geometric, as each term is double the previous term. Recognizing this pattern makes it easy to predict future terms by consistently applying the identified rule.
- A geometric pattern involves multiplying by a constant ratio, unlike an arithmetic pattern which involves addition.
- Patterns may not always be immediately obvious, requiring systematic checks (e.g., testing addition, multiplication).
Sequence Analysis
Analyzing sequences involves looking closely at the numbers and identifying the governing pattern. Sequence analysis helps in forecasting future terms and understanding the sequence structure.
In our exercise, the sequence 5, 10, 20, 40 grows by multiplying each term by 2. This defines a geometric progression with a common ratio of 2. By applying sequence analysis, one confirms this pattern and predicts additional terms like 80 and 160.
In our exercise, the sequence 5, 10, 20, 40 grows by multiplying each term by 2. This defines a geometric progression with a common ratio of 2. By applying sequence analysis, one confirms this pattern and predicts additional terms like 80 and 160.
- Start analysis by comparing differences or ratios between terms to identify sequence type.
- Test several hypotheses if the pattern is not clear. Patterns can involve various operations: addition, multiplication, or even alternating operations.
Other exercises in this chapter
Problem 16
Determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{2(-1)^{n+1}}{e^{n}+e^{-n}}=\sum_{n=1}^{\infty}(-1)^{n+1} \operatorname{sech}
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Use the power series $$\frac{1}{1+x}=\sum_{n=0}^{\infty}(-1)^{n} x^{n}$$ to determine a power series, centered at 0 , for the function. Identify the interval of
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Explain why the Integral Test does not apply to the series. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n}}{n} $$
View solution Problem 17
In Exercises \(7-18\), find the Maclaurin polynomial of degree \(n\) for the function. $$ f(x)=\sec x, \quad n=2 $$
View solution