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Chapter 4: Problem 7

Find the first five terms and the 50 th term of each infinite sequence defined. $$a_{n}=2 n-3$$

Short Answer

Expert verified
First five terms: -1, 1, 3, 5, 7. 50th term: 97.

Step by step solution

01

Identify the Formula for the Sequence

The given formula for the sequence is \( a_{n} = 2n - 3 \). This formula will help us find any term in the sequence based on its position \( n \).
02

Calculate the First Term

To find the first term (\( a_1 \)), substitute \( n = 1 \) into the formula: \[ a_1 = 2(1) - 3 = 2 - 3 = -1. \] Thus, the first term is \( -1 \).
03

Calculate the Second Term

To find the second term (\( a_2 \)), substitute \( n = 2 \) into the formula: \[ a_2 = 2(2) - 3 = 4 - 3 = 1. \] Thus, the second term is \( 1 \).
04

Calculate the Third Term

To find the third term (\( a_3 \)), substitute \( n = 3 \) into the formula: \[ a_3 = 2(3) - 3 = 6 - 3 = 3. \] Thus, the third term is \( 3 \).
05

Calculate the Fourth Term

To find the fourth term (\( a_4 \)), substitute \( n = 4 \) into the formula: \[ a_4 = 2(4) - 3 = 8 - 3 = 5. \] Thus, the fourth term is \( 5 \).
06

Calculate the Fifth Term

To find the fifth term (\( a_5 \)), substitute \( n = 5 \) into the formula: \[ a_5 = 2(5) - 3 = 10 - 3 = 7. \] Thus, the fifth term is \( 7 \).
07

Calculate the 50th Term

To find the 50th term (\( a_{50} \)), substitute \( n = 50 \) into the formula: \[ a_{50} = 2(50) - 3 = 100 - 3 = 97. \] Thus, the 50th term is \( 97 \).

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Key Concepts

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

Term Calculation
Calculating terms in an arithmetic sequence revolves around plugging different values into a given formula. For the sequence with the formula \( a_n = 2n - 3 \):
- Start by identifying the position \( n \) of the term you want to find.
- Substitute \( n \) into the formula.
- Perform the arithmetic operation: multiply \( 2 \) by \( n \) and then subtract \( 3 \).

This method allows you to calculate any term easily. For example, if you need to find the 10th term, substitute \( n = 10 \) in the formula: \( a_{10} = 2(10) - 3 = 17 \). This process underpins how arithmetic sequences operate, showing that each term is formed from a linear transformation of its position.
Sequence Formula
The sequence formula defines the pattern of the sequence. In our specific case, the formula is \( a_n = 2n - 3 \). This formula shows two vital components:
  • Linear growth: The term \( 2n \) indicates that as \( n \) increases, each term in the sequence increases linearly by 2. This regular increment ensures a consistent pattern.

  • Constant adjustment: The minus 3 adjusts each term by a fixed amount, shifting the entire sequence downward by 3 units.

Understanding the formula is key to predicting any term without calculating all the previous ones. Whether finding the 100th or the 8th term, this formula allows for direct computation, making long sequences manageable.
Mathematics Education
Teaching arithmetic sequences involves illustrating basic principles of linear relationships. It's crucial to:
  • Encourage students to recognize patterns in sequences via hands-on practice. Observing the incremental changes between terms reinforces understanding.

  • Explain the function of each part of a formula, like distinguishing between the multiplier and constant, to demystify the math involved.

  • Provide varied examples, such as increasing and decreasing sequences, to enhance comprehension.

This approach ensures that students grasp not only the how but also the why of arithmetic sequences, fostering more profound mathematical intuition. With engaging methods, students can build solid foundations, making complex math approachable in their educational journey.

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Most popular questions from this chapter

For arithmetic or geometric sequence. a) the 8th term b) an explicit formula for the \(n\)th term c) a recursive formula for the \(n\)th term. $$-2,3,-\frac{9}{2}, \frac{27}{4} \ldots$$

Suggest a recursive definition for each sequence. $$a-5 k, 2 a-4 k, 3 a-3 k, 4 a-2 k, 5 a-k, \ldots$$

Use your GDC or a spreadsheet to evaluate each sum. $$\sum_{k=1}^{20}\left(k^{2}+1\right)$$

Define the sequence $$F_{n}=\frac{1}{\sqrt{5}}\left(\frac{(1+\sqrt{5})^{n}-(1-\sqrt{5})^{n}}{2^{n}}\right)$$ a) Find the first 10 terms of this sequence and compare them to Fibonacci numbers. b) Show that \(3 \pm \sqrt{5}=\frac{(1 \pm \sqrt{5})^{2}}{2}\) c) Use the result in b) to verify that \(F_{n}\) satisfies the recursive definition of Fibonacci sequences.

Evaluate each expression. a) \(\left(\begin{array}{l}8 \\ 3\end{array}\right)\) b) \(\left(\begin{array}{c}18 \\ 5\end{array}\right)-\left(\begin{array}{c}18 \\\ 13\end{array}\right)\) c) \(\left(\begin{array}{l}7 \\ 4\end{array}\right)\left(\begin{array}{l}7 \\\ 3\end{array}\right)\) d) \(\left(\begin{array}{l}5 \\ 0\end{array}\right)+\left(\begin{array}{l}5 \\\ 1\end{array}\right)+\left(\begin{array}{l}5 \\\ 2\end{array}\right)+\left(\begin{array}{l}5 \\\ 3\end{array}\right)+\left(\begin{array}{l}5 \\\ 4\end{array}\right)+\left(\begin{array}{l}5 \\ 5\end{array}\right)\) e) \(\left(\begin{array}{l}6 \\ 0\end{array}\right)-\left(\begin{array}{l}6 \\\ 1\end{array}\right)+\left(\begin{array}{l}6 \\\ 2\end{array}\right)-\left(\begin{array}{l}6 \\\ 3\end{array}\right)+\left(\begin{array}{l}6 \\\ 4\end{array}\right)-\left(\begin{array}{l}6 \\\ 5\end{array}\right)+\left(\begin{array}{l}6 \\ 6\end{array}\right)\)
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