Question
a realtor sold 50 houses in her first year and plans to increase her sales by 15% each year for the next 8 years how many houses will she have sold at the end of her 8th year
identify Aunder1 n and r
make sure you write the equation to find the sum
show all your work
correct sum
identify Aunder1 n and r
make sure you write the equation to find the sum
show all your work
correct sum
Answers
Answered by
GPT-4o mini
To determine how many houses the realtor will have sold at the end of her 8th year, we can use the formula for geometric series since she plans to increase her sales by a constant percentage each year.
### Step 1: Identify the components
1. **Initial sales (A)**: The number of houses sold in the first year is 50.
2. **Growth rate (r)**: The sales increase by 15% each year, which means \( r = 1 + 0.15 = 1.15 \).
3. **Number of years (n)**: She plans to continue this for 8 years, but we will calculate the total sold after 8 years.
### Step 2: Sales for each year
The number of houses sold in each year can be defined using the formula for a geometric sequence:
\[
A_n = A \cdot r^{n-1}
\]
Where:
- \( A_n \) is the number of houses sold in year n,
- \( A \) is the initial amount (50),
- \( r \) is the growth factor (1.15),
- \( n \) is the year number.
### Step 3: Calculating the sales for each year
We will calculate the total number of houses sold over the 8 years. The total number sold over these years can be expressed as:
\[
S_n = A \frac{(r^n - 1)}{(r - 1)}
\]
Where:
- \( S_n \) is the total number of houses sold after n years,
- \( n \) = 8 (because we want the total after 8 years).
### Step 4: Plug in the values into the sum formula
Now let's calculate it:
- \( A = 50 \)
- \( r = 1.15 \)
- \( n = 8 \)
Putting these values into the formula for the sum of the geometric series:
\[
S_8 = 50 \cdot \frac{(1.15^8 - 1)}{(1.15 - 1)}
\]
Now calculating \( 1.15^8 \):
\[
1.15^8 \approx 3.4323
\]
Now substitute this value back into the formula:
\[
S_8 = 50 \cdot \frac{(3.4323 - 1)}{(0.15)}
\]
Calculate \( 3.4323 - 1 = 2.4323 \):
\[
S_8 = 50 \cdot \frac{2.4323}{0.15}
\]
Calculating \( \frac{2.4323}{0.15} \):
\[
\frac{2.4323}{0.15} \approx 16.2153
\]
Now, multiply by 50:
\[
S_8 \approx 50 \cdot 16.2153 = 810.765
\]
### Final result
Since the number of houses sold should be a whole number, we round it to the nearest whole number:
The number of houses sold at the end of her 8th year is approximately **811 houses**.
Thus, the correct sum of houses sold over 8 years is **811**.
### Step 1: Identify the components
1. **Initial sales (A)**: The number of houses sold in the first year is 50.
2. **Growth rate (r)**: The sales increase by 15% each year, which means \( r = 1 + 0.15 = 1.15 \).
3. **Number of years (n)**: She plans to continue this for 8 years, but we will calculate the total sold after 8 years.
### Step 2: Sales for each year
The number of houses sold in each year can be defined using the formula for a geometric sequence:
\[
A_n = A \cdot r^{n-1}
\]
Where:
- \( A_n \) is the number of houses sold in year n,
- \( A \) is the initial amount (50),
- \( r \) is the growth factor (1.15),
- \( n \) is the year number.
### Step 3: Calculating the sales for each year
We will calculate the total number of houses sold over the 8 years. The total number sold over these years can be expressed as:
\[
S_n = A \frac{(r^n - 1)}{(r - 1)}
\]
Where:
- \( S_n \) is the total number of houses sold after n years,
- \( n \) = 8 (because we want the total after 8 years).
### Step 4: Plug in the values into the sum formula
Now let's calculate it:
- \( A = 50 \)
- \( r = 1.15 \)
- \( n = 8 \)
Putting these values into the formula for the sum of the geometric series:
\[
S_8 = 50 \cdot \frac{(1.15^8 - 1)}{(1.15 - 1)}
\]
Now calculating \( 1.15^8 \):
\[
1.15^8 \approx 3.4323
\]
Now substitute this value back into the formula:
\[
S_8 = 50 \cdot \frac{(3.4323 - 1)}{(0.15)}
\]
Calculate \( 3.4323 - 1 = 2.4323 \):
\[
S_8 = 50 \cdot \frac{2.4323}{0.15}
\]
Calculating \( \frac{2.4323}{0.15} \):
\[
\frac{2.4323}{0.15} \approx 16.2153
\]
Now, multiply by 50:
\[
S_8 \approx 50 \cdot 16.2153 = 810.765
\]
### Final result
Since the number of houses sold should be a whole number, we round it to the nearest whole number:
The number of houses sold at the end of her 8th year is approximately **811 houses**.
Thus, the correct sum of houses sold over 8 years is **811**.