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Chapter 10: Problem 1

Write the first five terms of the sequence. $$ a_{n}=2 n-1 $$

Short Answer

Expert verified
The first five terms of the sequence are 1, 3, 5, 7 and 9.

Step by step solution

01

- Determine the first term

Plug n = 1 into the formula to find the first term: \( a_1 = 2(1) - 1 = 1 \)
02

- Determine the second term

Plug n = 2 into the formula to find the second term: \(a_2 = 2(2) - 1 = 3 \)
03

- Determine the third term

Plug n = 3 into the formula to find the third term: \(a_3 = 2(3) - 1 = 5 \)
04

- Determine the fourth term

Plug n = 4 into the formula to find the fourth term: \(a_4 = 2(4) - 1 = 7 \)
05

- Determine the fifth term

Plug n = 5 into the formula to find the fifth term: \(a_5 = 2(5) - 1 = 9 \)

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Key Concepts

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

Sequence Formula
In arithmetic sequences, the sequence formula is fundamental. It is the rule which helps us find any term in the sequence without listing all terms. The given sequence formula is \( a_n = 2n - 1 \).
This formula shows that every term, \( a_n \), in the sequence can be found using its position number, \( n \).
For any positive integer \( n \), you simply plug it into the formula.
  • The \( 2n \) part implies that the sequence increments by 2 each step.
  • The \(-1\) part adjusts each term to reflect the sequence's starting point.
This formula is highly useful because it allows calculating higher-order terms without computing all preceding terms.
Term Calculation
Term calculation in arithmetic sequences involves using the sequence formula to find specific terms. Let's break down how to calculate the first five terms using the given formula.
1. **First Term:** By inserting \( n = 1 \) into the formula, \( a_1 = 2(1)-1 = 1 \).
2. **Second Term:** Substitute \( n = 2 \) to get \( a_2 = 2(2)-1 = 3 \).
3. **Third Term:** With \( n = 3 \), you find \( a_3 = 2(3)-1 = 5 \).
4. **Fourth Term:** By setting \( n = 4 \), the term becomes \( a_4 = 2(4)-1 = 7 \).
5. **Fifth Term:** For \( n = 5 \), the term is \( a_5 = 2(5)-1 = 9 \).
  • Each calculation follows the same process: plug \( n \) into \( a_n = 2n-1 \).
  • This consistency simplifies finding any term you need.
Linear Patterns
Arithmetic sequences are characterized by linear patterns. Each term increases by a fixed amount from the previous one, reflecting a constant difference.
In our formula \( a_n = 2n - 1 \), the sequence generates: 1, 3, 5, 7, 9, revealing a steady step of 2.
Recognizing this pattern is critical for predicting future terms and understanding how arithmetic sequences function.
  • The pattern of adding 2 each time means the sequence is a linear progression.
  • Arithmetic sequences differ from geometric ones, where each term is multiplied by a constant.
This predictability is a key feature, making arithmetic sequences essential in various fields like finance and computer science.

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Most popular questions from this chapter

Write the next two terms of the geometric sequence. Describe the pattern you used to find these terms. $$ 5,10,20,40, \dots $$

Verify that the infinite series diverges. $$ \sum_{n=1}^{\infty} \frac{n}{n+1}=\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\frac{4}{5}+\cdots $$

Salary You go to work at a company that pays 0.01 dollars for the first day, 0.02 dollars for the second day, 0.04 dollars for the third day, and so on. If the daily wage keeps doubling, what would your total income be for working (a) 29 days, (b) 30 days, and (c) 31 days?

Write the first five terms of the sequence of partial sums. $$ \sum_{n=1}^{\infty} \frac{3}{2^{n-1}}=3+\frac{3}{2}+\frac{3}{4}+\frac{3}{8}+\frac{3}{16}+\cdots $$

Verify that the infinite series diverges. $$ \sum_{n=0}^{\infty} 3\left(\frac{3}{2}\right)^{n}=3+\frac{9}{2}+\frac{27}{4}+\frac{81}{8}+\cdots $$
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