If The Nth Term Of A Series Is $3^n-2^{n+1}$, Find The Difference Between The $7^{\text{th}}$ And \$9^{\text{th}}$[/tex\] Terms.
If the nth term of a series is $3n-2{n+1}$, find the difference between the $7^{\text{th}}$ and $$ terms
In mathematics, a series is a sequence of numbers in which each term is obtained by a specific rule. The nth term of a series is the term that appears at the nth position in the sequence. In this article, we will explore the concept of the nth term of a series and use it to find the difference between the 7th and 9th terms of a given series.
The nth term of a series is denoted by the formula $a_n = a_1 + (n-1)d$, where $a_1$ is the first term of the series, $d$ is the common difference between consecutive terms, and $n$ is the position of the term in the series. However, in this problem, we are given a specific formula for the nth term of the series: $3n-2{n+1}$.
Understanding the Formula
To understand the formula, let's break it down into its components. The formula consists of two parts: $3^n$ and $-2^{n+1}$. The first part, $3^n$, represents the nth power of 3, which means that the base number 3 is raised to the power of n. The second part, $-2^{n+1}$, represents the negative of the (n+1)th power of 2.
Calculating the 7th Term
To calculate the 7th term of the series, we need to substitute n = 7 into the formula: $37-2{7+1}$. This simplifies to:
Using a calculator, we can evaluate this expression as follows:
Therefore, the 7th term of the series is:
Calculating the 9th Term
To calculate the 9th term of the series, we need to substitute n = 9 into the formula: $39-2{9+1}$. This simplifies to:
Using a calculator, we can evaluate this expression as follows:
Therefore, the 9th term of the series is:
Finding the Difference
To find the difference between the 7th and 9th terms of the series, we need to subtract the 7th term from the 9th term:
Therefore, the difference between the 7th and 9th terms of the series is 16728.
In this article, we used the formula for the nth term of a series to find the difference between the 7th and 9th terms of a given series. We calculated the 7th and 9th terms of the series using the formula and then found the difference between them. The result is 16728.
The final answer is:
If the nth term of a series is $3n-2{n+1}$, find the difference between the $7^{\text{th}}$ and $$ terms
In mathematics, a series is a sequence of numbers in which each term is obtained by a specific rule. The nth term of a series is the term that appears at the nth position in the sequence. In this article, we will explore the concept of the nth term of a series and use it to find the difference between the 7th and 9th terms of a given series.
The nth term of a series is denoted by the formula $a_n = a_1 + (n-1)d$, where $a_1$ is the first term of the series, $d$ is the common difference between consecutive terms, and $n$ is the position of the term in the series. However, in this problem, we are given a specific formula for the nth term of the series: $3n-2{n+1}$.
Understanding the Formula
To understand the formula, let's break it down into its components. The formula consists of two parts: $3^n$ and $-2^{n+1}$. The first part, $3^n$, represents the nth power of 3, which means that the base number 3 is raised to the power of n. The second part, $-2^{n+1}$, represents the negative of the (n+1)th power of 2.
Calculating the 7th Term
To calculate the 7th term of the series, we need to substitute n = 7 into the formula: $37-2{7+1}$. This simplifies to:
Using a calculator, we can evaluate this expression as follows:
Therefore, the 7th term of the series is:
Calculating the 9th Term
To calculate the 9th term of the series, we need to substitute n = 9 into the formula: $39-2{9+1}$. This simplifies to:
Using a calculator, we can evaluate this expression as follows:
Therefore, the 9th term of the series is:
Finding the Difference
To find the difference between the 7th and 9th terms of the series, we need to subtract the 7th term from the 9th term:
Therefore, the difference between the 7th and 9th terms of the series is 16728.
Q: What is the nth term of a series? A: The nth term of a series is the term that appears at the nth position in the sequence.
Q: How do you calculate the nth term of a series? A: To calculate the nth term of a series, you need to substitute n into the formula for the nth term.
Q: What is the formula for the nth term of the series in this problem? A: The formula for the nth term of the series in this problem is $3n-2{n+1}$.
Q: How do you find the difference between the 7th and 9th terms of the series? A: To find the difference between the 7th and 9th terms of the series, you need to subtract the 7th term from the 9th term.
Q: What is the difference between the 7th and 9th terms of the series? A: The difference between the 7th and 9th terms of the series is 16728.
Q: Can you explain the concept of the nth term of a series in more detail? A: The nth term of a series is a specific term in a sequence of numbers. It is obtained by substituting n into the formula for the nth term. The formula for the nth term of a series can be used to calculate any term in the series.
Q: How do you use the formula for the nth term of a series to find the difference between two terms? A: To use the formula for the nth term of a series to find the difference between two terms, you need to substitute the values of n into the formula and then subtract the two terms.
In this article, we used the formula for the nth term of a series to find the difference between the 7th and 9th terms of a given series. We calculated the 7th and 9th terms of the series using the formula and then found the difference between them. The result is 16728.
The final answer is: