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Answer :
- Establish equations for the 3rd and 9th terms of the AP: $a + 2d = 4$ and $a + 8d = -8$.
- Solve the system of equations to find $a = 8$ and $d = -2$.
- Use the formula for the nth term, $a_n = a + (n-1)d$, and set $a_n = 0$.
- Solve for $n$ to find that the 5th term is zero: $\boxed{5}$.
### Explanation
1. Understanding the Problem
We are given that the 3rd term of an arithmetic progression (AP) is 4 and the 9th term is -8. Our goal is to find which term of this AP is equal to zero. Let's denote the first term of the AP as $a$ and the common difference as $d$.
2. Formulating Equations
The $n^{th}$ term of an AP is given by $a_n = a + (n-1)d$. Using the given information, we can write two equations:
For the 3rd term ($n=3$):
$$a_3 = a + (3-1)d = a + 2d = 4$$
For the 9th term ($n=9$):
$$a_9 = a + (9-1)d = a + 8d = -8$$
3. Solving for the Common Difference
Now we have a system of two linear equations with two variables, $a$ and $d$:
$$a + 2d = 4$$
$$a + 8d = -8$$
We can solve this system using substitution or elimination. Let's use elimination. Subtract the first equation from the second equation:
$$(a + 8d) - (a + 2d) = -8 - 4$$
$$6d = -12$$
$$d = -2$$
4. Solving for the First Term
Now that we have the value of $d$, we can substitute it back into either of the original equations to find the value of $a$. Let's use the first equation:
$$a + 2d = 4$$
$$a + 2(-2) = 4$$
$$a - 4 = 4$$
$$a = 8$$
5. Finding the Term that is Zero
Now we have the first term $a = 8$ and the common difference $d = -2$. We want to find the term $n$ for which $a_n = 0$. Using the formula for the $n^{th}$ term:
$$a_n = a + (n-1)d$$
$$0 = 8 + (n-1)(-2)$$
$$0 = 8 - 2n + 2$$
$$0 = 10 - 2n$$
$$2n = 10$$
$$n = 5$$
6. Conclusion
Therefore, the 5th term of the AP is zero.
### Examples
Arithmetic progressions are useful in various real-life scenarios such as calculating simple interest, predicting salary increases, and determining the number of seats in an auditorium. For example, if you deposit a fixed amount of money into a savings account each month, the total amount in your account over time forms an arithmetic progression. Understanding APs can help you predict future values and plan accordingly.
- Solve the system of equations to find $a = 8$ and $d = -2$.
- Use the formula for the nth term, $a_n = a + (n-1)d$, and set $a_n = 0$.
- Solve for $n$ to find that the 5th term is zero: $\boxed{5}$.
### Explanation
1. Understanding the Problem
We are given that the 3rd term of an arithmetic progression (AP) is 4 and the 9th term is -8. Our goal is to find which term of this AP is equal to zero. Let's denote the first term of the AP as $a$ and the common difference as $d$.
2. Formulating Equations
The $n^{th}$ term of an AP is given by $a_n = a + (n-1)d$. Using the given information, we can write two equations:
For the 3rd term ($n=3$):
$$a_3 = a + (3-1)d = a + 2d = 4$$
For the 9th term ($n=9$):
$$a_9 = a + (9-1)d = a + 8d = -8$$
3. Solving for the Common Difference
Now we have a system of two linear equations with two variables, $a$ and $d$:
$$a + 2d = 4$$
$$a + 8d = -8$$
We can solve this system using substitution or elimination. Let's use elimination. Subtract the first equation from the second equation:
$$(a + 8d) - (a + 2d) = -8 - 4$$
$$6d = -12$$
$$d = -2$$
4. Solving for the First Term
Now that we have the value of $d$, we can substitute it back into either of the original equations to find the value of $a$. Let's use the first equation:
$$a + 2d = 4$$
$$a + 2(-2) = 4$$
$$a - 4 = 4$$
$$a = 8$$
5. Finding the Term that is Zero
Now we have the first term $a = 8$ and the common difference $d = -2$. We want to find the term $n$ for which $a_n = 0$. Using the formula for the $n^{th}$ term:
$$a_n = a + (n-1)d$$
$$0 = 8 + (n-1)(-2)$$
$$0 = 8 - 2n + 2$$
$$0 = 10 - 2n$$
$$2n = 10$$
$$n = 5$$
6. Conclusion
Therefore, the 5th term of the AP is zero.
### Examples
Arithmetic progressions are useful in various real-life scenarios such as calculating simple interest, predicting salary increases, and determining the number of seats in an auditorium. For example, if you deposit a fixed amount of money into a savings account each month, the total amount in your account over time forms an arithmetic progression. Understanding APs can help you predict future values and plan accordingly.
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