We appreciate your visit to 11 Which term of the arithmetic progression AP tex 3 15 27 39 tex will be 120 more than its 21st term 12 The 8th. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!
Answer :
- Determine the first term $a = 3$ and common difference $d = 12$.
- Calculate the 21st term: $a_{21} = 3 + 20(12) = 243$.
- Set up the equation $a_n = a_{21} + 120 = 363$.
- Solve for $n$: $363 = 3 + (n-1)12$, which gives $n = 31$. The final answer is $\boxed{31}$.
### Explanation
1. Problem Analysis
The arithmetic progression (AP) is given by $3, 15, 27, 39, ...$. We need to find which term of this AP is 120 more than its 21st term.
2. Finding First Term and Common Difference
First, let's find the first term ($a$) and the common difference ($d$) of the AP.
The first term, $a$, is 3.
The common difference, $d$, is the difference between consecutive terms: $15 - 3 = 12$. So, $d = 12$.
3. Setting up the Equation
The $n^{th}$ term of an AP is given by the formula: $a_n = a + (n-1)d$.
We are given that the $n^{th}$ term is 120 more than the 21st term. So, $a_n = a_{21} + 120$.
4. Calculating the 21st Term
Now, let's find the 21st term, $a_{21}$:
$a_{21} = a + (21-1)d = a + 20d = 3 + 20(12) = 3 + 240 = 243$.
5. Finding the Term Number
We have $a_n = a_{21} + 120$, so $a_n = 243 + 120 = 363$.
Now, we need to find the value of $n$ for which $a_n = 363$.
Using the formula for the $n^{th}$ term: $a_n = a + (n-1)d$, we have:
$363 = 3 + (n-1)12$
$360 = (n-1)12$
$30 = n-1$
$n = 31$
6. Final Answer
Therefore, the 31st term of the AP is 120 more than its 21st term.
### Examples
Arithmetic progressions are useful in various real-life scenarios, such as calculating simple interest, predicting salary increases, or determining the number of seats in rows of a theater. Understanding how to find a specific term or the difference between terms can help in financial planning, resource allocation, and making informed decisions based on sequential data.
- Calculate the 21st term: $a_{21} = 3 + 20(12) = 243$.
- Set up the equation $a_n = a_{21} + 120 = 363$.
- Solve for $n$: $363 = 3 + (n-1)12$, which gives $n = 31$. The final answer is $\boxed{31}$.
### Explanation
1. Problem Analysis
The arithmetic progression (AP) is given by $3, 15, 27, 39, ...$. We need to find which term of this AP is 120 more than its 21st term.
2. Finding First Term and Common Difference
First, let's find the first term ($a$) and the common difference ($d$) of the AP.
The first term, $a$, is 3.
The common difference, $d$, is the difference between consecutive terms: $15 - 3 = 12$. So, $d = 12$.
3. Setting up the Equation
The $n^{th}$ term of an AP is given by the formula: $a_n = a + (n-1)d$.
We are given that the $n^{th}$ term is 120 more than the 21st term. So, $a_n = a_{21} + 120$.
4. Calculating the 21st Term
Now, let's find the 21st term, $a_{21}$:
$a_{21} = a + (21-1)d = a + 20d = 3 + 20(12) = 3 + 240 = 243$.
5. Finding the Term Number
We have $a_n = a_{21} + 120$, so $a_n = 243 + 120 = 363$.
Now, we need to find the value of $n$ for which $a_n = 363$.
Using the formula for the $n^{th}$ term: $a_n = a + (n-1)d$, we have:
$363 = 3 + (n-1)12$
$360 = (n-1)12$
$30 = n-1$
$n = 31$
6. Final Answer
Therefore, the 31st term of the AP is 120 more than its 21st term.
### Examples
Arithmetic progressions are useful in various real-life scenarios, such as calculating simple interest, predicting salary increases, or determining the number of seats in rows of a theater. Understanding how to find a specific term or the difference between terms can help in financial planning, resource allocation, and making informed decisions based on sequential data.
Thanks for taking the time to read 11 Which term of the arithmetic progression AP tex 3 15 27 39 tex will be 120 more than its 21st term 12 The 8th. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!
- Why do Businesses Exist Why does Starbucks Exist What Service does Starbucks Provide Really what is their product.
- The pattern of numbers below is an arithmetic sequence tex 14 24 34 44 54 ldots tex Which statement describes the recursive function used to..
- Morgan felt the need to streamline Edison Electric What changes did Morgan make.
Rewritten by : Barada
