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Answer :
The problem involves finding the value of $n$ for which the $n$th terms of two arithmetic progressions are equal.
- Define the parameters of both arithmetic progressions.
- Express the $n$th term of each AP using the formula $a_n = a + (n-1)d$.
- Set the expressions for the $n$th terms equal to each other and solve for $n$.
- The $n$th terms are equal when $n = \boxed{13}$.
### Explanation
1. Understanding the Problem
We are given two arithmetic progressions (APs): $63, 65, 67, \ldots$ and $3, 10, 17, \ldots$. Our goal is to find the value of $n$ for which the $n$th terms of both APs are equal.
2. Identifying the AP parameters
Let's denote the first AP as AP1 and the second as AP2. For AP1, the first term is $a_1 = 63$ and the common difference is $d_1 = 65 - 63 = 2$. For AP2, the first term is $a_2 = 3$ and the common difference is $d_2 = 10 - 3 = 7$.
3. Recalling the formula for the nth term
The $n$th term of an AP is given by the formula $a_n = a + (n-1)d$, where $a$ is the first term and $d$ is the common difference. We want to find $n$ such that the $n$th term of AP1 equals the $n$th term of AP2.
4. Finding the general formulas
The $n$th term of AP1 is $a_{1n} = a_1 + (n-1)d_1 = 63 + (n-1)2$. The $n$th term of AP2 is $a_{2n} = a_2 + (n-1)d_2 = 3 + (n-1)7$.
5. Equating the nth terms
Now, we set the two $n$th terms equal to each other: $$63 + (n-1)2 = 3 + (n-1)7$$
6. Solving for n
Let's solve the equation for $n$:
$$63 + 2n - 2 = 3 + 7n - 7$$
$$61 + 2n = 7n - 4$$
$$65 = 5n$$
$$n = \frac{65}{5} = 13$$
7. Final Answer
Therefore, the $n$th terms of the two APs are equal when $n = 13$.
### Examples
Arithmetic progressions are useful in various real-life scenarios, such as calculating simple interest on a loan or predicting the growth of a population over time. In this case, finding the value of 'n' where two APs have the same term can be applied to scenarios like comparing investment plans with different initial values and growth rates to determine when they yield the same return.
- Define the parameters of both arithmetic progressions.
- Express the $n$th term of each AP using the formula $a_n = a + (n-1)d$.
- Set the expressions for the $n$th terms equal to each other and solve for $n$.
- The $n$th terms are equal when $n = \boxed{13}$.
### Explanation
1. Understanding the Problem
We are given two arithmetic progressions (APs): $63, 65, 67, \ldots$ and $3, 10, 17, \ldots$. Our goal is to find the value of $n$ for which the $n$th terms of both APs are equal.
2. Identifying the AP parameters
Let's denote the first AP as AP1 and the second as AP2. For AP1, the first term is $a_1 = 63$ and the common difference is $d_1 = 65 - 63 = 2$. For AP2, the first term is $a_2 = 3$ and the common difference is $d_2 = 10 - 3 = 7$.
3. Recalling the formula for the nth term
The $n$th term of an AP is given by the formula $a_n = a + (n-1)d$, where $a$ is the first term and $d$ is the common difference. We want to find $n$ such that the $n$th term of AP1 equals the $n$th term of AP2.
4. Finding the general formulas
The $n$th term of AP1 is $a_{1n} = a_1 + (n-1)d_1 = 63 + (n-1)2$. The $n$th term of AP2 is $a_{2n} = a_2 + (n-1)d_2 = 3 + (n-1)7$.
5. Equating the nth terms
Now, we set the two $n$th terms equal to each other: $$63 + (n-1)2 = 3 + (n-1)7$$
6. Solving for n
Let's solve the equation for $n$:
$$63 + 2n - 2 = 3 + 7n - 7$$
$$61 + 2n = 7n - 4$$
$$65 = 5n$$
$$n = \frac{65}{5} = 13$$
7. Final Answer
Therefore, the $n$th terms of the two APs are equal when $n = 13$.
### Examples
Arithmetic progressions are useful in various real-life scenarios, such as calculating simple interest on a loan or predicting the growth of a population over time. In this case, finding the value of 'n' where two APs have the same term can be applied to scenarios like comparing investment plans with different initial values and growth rates to determine when they yield the same return.
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