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.
