Problem 23

Question

Determine the common difference, the fifth term, the $n$ th term, and the 100 th term of the arithmetic sequence. $$2,5,8,11, \dots$$

Step-by-Step Solution

Verified

Answer

The common difference is 3, the 5th term is 14, the nth term is $3n-1$, and the 100th term is 299.

1Step 1: Identify the Common Difference

To find the common difference in an arithmetic sequence, subtract the first term from the second term. Here, the terms are 2, 5, 8, and 11. The common difference $d$ is calculated as follows: $d = 5 - 2 = 3$.

2Step 2: Calculate the Fifth Term

The formula for the nth term of an arithmetic sequence is $a_n = a_1 + (n-1) imes d$. We need to find the 5th term $a_5$. Substituting into the formula: $a_5 = 2 + (5-1) imes 3 = 2 + 12 = 14$.

3Step 3: Derive the General Formula for the nth Term

Using the nth term formula, $a_n = a_1 + (n-1) imes d$, where $a_1 = 2$ and $d = 3$, the general formula is: $a_n = 2 + (n-1) imes 3$. Simplify it to: $a_n = 3n - 1$.

4Step 4: Calculate the 100th Term

Using the nth term formula derived in Step 3, $a_n = 3n - 1$, find the 100th term $a_{100}$. Substitute $n = 100$ into the equation: $a_{100} = 3 imes 100 - 1 = 300 - 1 = 299$.

Key Concepts

Understanding the Common DifferenceDemystifying the nth Term FormulaCalculating the General Term

Understanding the Common Difference

In an arithmetic sequence, the common difference is the key to understanding how the sequence progresses. It is the constant difference between consecutive terms in the sequence. In simple terms, it's how much we "jump" from one number to the next.
For example, in the sequence given: 2, 5, 8, 11, ..., to find this common difference, you subtract the first term from the second term. Here it is calculated as follows:

5 - 2 = 3

This tells us that each term is 3 more than the previous one. Knowing this common difference helps in predicting and calculating other terms in the sequence easily.

Demystifying the nth Term Formula

The nth term formula in an arithmetic sequence lets you calculate any term in the sequence without listing all other terms. It is expressed as:

$a_n = a_1 + (n-1) \times d $

Here, $a_n$ is the term you want to find, $a_1$ is the first term, $d$ is the common difference, and $n$ is the term number you're solving for.
For instance, to find the fifth term using this sequence, plug the values into the formula:

$a_5 = 2 + (5-1) \times 3 = 14$

This formula is versatile, meaning you can find any term, not just the initial ones. Just substitute the desired term number for $n$ and you’re set!

Calculating the General Term

The general term of an arithmetic sequence provides a simple formula to solve for any term without having to go step-by-step. This helps in efficiently working with large sequences.
Using the sequence provided, we derived the general formula:

$a_n = 3n - 1$

Here, the formula comes from simplifying $a_n = 2 + (n-1) \times 3$, which is achieved through distribution of the common difference.For an application, to find the 100th term of the sequence, substitute 100 for $n$ in the formula:

$a_{100} = 3 \times 100 - 1 = 299$

This way, the general term formula simplifies determining any term, providing both a quick and systematic approach.

Problem 22

Problem 23

Other exercises in this chapter

Problem 22

Show that $100 n \leq n^{2}$ for all $n \geq 100$

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Problem 22

Use a graphing calculator to do the following. (a) Find the first 10 terms of the sequence. (b) Graph the first 10 terms of the sequence. $$a_{n}=a_{n-1}-a_{n-2

View solution

Problem 23

Use the Binomial Theorcm to expand the expression. $$\left(1+\frac{1}{x}\right)^{6}$$

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Problem 23

Let $a_{n+1}=3 a_{n}$ and $a_{1}=5 .$ Show that $a_{n}=5 \cdot 3^{n-1}$ for all natural numbers $n$

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