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Chapter 10: Problem 6

Write the first four terms of each sequence. Assume \(n\) starts at 1. $$a_{n}=\frac{(n+1)}{n}$$

Short Answer

Expert verified
The first four terms are 2, \( \frac{3}{2} \), \( \frac{4}{3} \), and \( \frac{5}{4} \).

Step by step solution

01

Identify the formula

The sequence is given by the formula \( a_n = \frac{n+1}{n} \). We'll use this formula to find the terms of the sequence by substituting different values of \( n \).
02

Calculate the first term

Substitute \( n = 1 \) into the given formula.\[ a_1 = \frac{1+1}{1} = \frac{2}{1} = 2 \]So, the first term \( a_1 \) is 2.
03

Calculate the second term

Substitute \( n = 2 \) into the formula.\[ a_2 = \frac{2+1}{2} = \frac{3}{2} \]Therefore, the second term \( a_2 \) is \( \frac{3}{2} \).
04

Calculate the third term

Substitute \( n = 3 \) into the formula.\[ a_3 = \frac{3+1}{3} = \frac{4}{3} \]Thus, the third term \( a_3 \) is \( \frac{4}{3} \).
05

Calculate the fourth term

Substitute \( n = 4 \) into the formula.\[ a_4 = \frac{4+1}{4} = \frac{5}{4} \]Hence, the fourth term \( a_4 \) is \( \frac{5}{4} \).

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Key Concepts

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

Terms of a Sequence
In mathematics, a sequence is an ordered list of numbers where each number is called a term. Sequences can be arithmetic, geometric, or follow any general pattern defined by a rule.
To understand and solve problems involving sequences, especially as they appear in math homework or exams, becoming familiar with terms is crucial. Let's dive into what the terms mean:
  • The position of a term in a sequence is often represented by a variable, commonly denoted as \( n \).
  • In the sequence formula provided in this exercise, like \( a_n = \frac{n+1}{n} \), each \( a_n \) is a specific term corresponding to a particular position \( n \).
  • The task often involves finding the first few terms, which means identifying the values for the positions starting from \( n = 1 \).
By finding the terms one by one, math students knock down a sequence challenge by understanding how each position relates to the following one.
Substitution Method
The substitution method is a key technique used to find terms of a sequence. It's straightforward鈥攕ubstitute different specific values into an expression or equation. This method helps in discovering the values of the sequence step by step. Here's how you can apply it to sequences:
  • Take the given sequence formula, such as \( a_n = \frac{n+1}{n} \).
  • Pick a value for \( n \), starting typically with 1, since sequences often start from the first term.
  • Replace \( n \) in your formula with that number. For example, substituting \( n = 1 \) would give \( a_1 = \frac{1+1}{1} \), which simplifies to 2.
  • Repeat this process for the next few integers (i.e., \( n = 2, 3, 4\), etc.).
Ultimately, substitution enables a stepwise calculation of sequence terms, confirming both understanding and accuracy.
Sequence Formula
The sequence formula is the backbone of many sequence problems in mathematics. It defines the pattern of progression that allows us to calculate any term in a sequence. Essentially, it's like a recipe for the terms. Here's what you should know:
  • A sequence formula expresses the rule for the sequence, providing a general description of how terms are formed.
  • In many standard exercises, you're given a formula, such as \( a_n = \frac{n+1}{n} \), which generates each number in the list from a specific \( n \) value.
  • This formula often involves operations on \( n \), and each operation plays a critical part in forming each term.
Understanding and applying the sequence formula requires recognizing the pattern the formula describes, ensuring you can solve exercises related to any sequence given. It's your guide through the mathematical maze, showing exactly how to derive each term.

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Most popular questions from this chapter

Write the first four terms of the sequence defined by each recursion formula. Assume the sequence begins at \(n=1\). $$a_{1}=1, a_{2}=2 \quad a_{n}=a_{n-1} \cdot a_{n-2}$$

Evaluate each finite series. $$\sum_{k=0}^{5} \frac{2^{k}}{k !}$$

Use a graphing calculator to find \(\sum_{n=1}^{200}\left[-18+\frac{4}{5}(n-1)\right]\).

The Fibonacci sequence is defined by \(a_{1}=1, a_{2}=1,\) and \(a_{n}=a_{n-2}+a_{n-1}\) for \(n \geq 3 .\) The ratio \(\frac{a_{n+1}}{a_{n}}\) is an approximation of the golden ratio. The ratio approaches a constant \(\phi\) (phi) as \(n\) gets large. Find the golden ratio using a graphing utility.

In calculus, when estimating certain integrals, we use sums of the form \(\sum_{i=1}^{n} f\left(x_{i}\right) \Delta x,\) where \(f\) is a function and \(\Delta x\) is a constant. Find the indicated sum. $$\sum_{i=1}^{43} f\left(x_{i}\right) \Delta x, \text { where } f\left(x_{i}\right)=6+i \text { and } \Delta x=0.001$$
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