A gardener is making a planting in the shape of a trapezoid. It will have 41 plants in the front row, 37 in the second row, 33 in the third row, and so on. If the pattern is consistent, how many plants will there be in the last row? How many plants are there altogether?
Thank you!
The nth row,
Rn = 41-4(n-1) = 45-4n
So, there are 11 rows, since 45-48 < 0
You have an AP with
a = 41
d = -4
S11 = 11/2 (2*41 + 10(-4)) = 231
To find the number of plants in the last row, we need to determine the pattern and find a way to express it mathematically. Let's observe the number of plants in each row:
Row 1: 41 plants
Row 2: 37 plants
Row 3: 33 plants
We can see that the number of plants is decreasing by 4 with each row. So, the pattern can be expressed as:
Number of plants in each row = 41 - 4 * (row number - 1)
To find the number of plants in the last row, we need to determine the value of the row number. Since the pattern is consistent, we can derive the formula to find the row number:
(row number) = (first row number) + (number of rows - 1)
In this case, the first row number is 1, and there are a total of 41 rows. Substituting the values into the formula, we have:
(row number) = 1 + (41 - 1) = 1 + 40 = 41
Now, let's find the number of plants in the last row using the formula we derived earlier:
Number of plants in the last row = 41 - 4 * (41 - 1) = 41 - 4 * 40 = 41 - 160 = -119
However, since negative numbers of plants are not possible, we can conclude that the last row will not have any plants.
To find the total number of plants, we can use the sum of an arithmetic series formula:
Total number of plants = (number of rows / 2) * (first row number + last row number)
In this case, the number of rows is 41, and the first row number is 41. Since the last row has no plants, the last row number is considered as 0. Substituting the values into the formula, we have:
Total number of plants = (41 / 2) * (41 + 0) = 20.5 * 41 = 841
Therefore, there are a total of 841 plants in the planting.
To determine how many plants will be in the last row and the total number of plants, we need to identify the pattern in the number of plants in each row.
We can observe that the number of plants in each row decreases by 4 from the previous row. So, we need to find the number of rows until we reach 0 or a negative number.
To find the number of rows, we can use the formula for the sum of an arithmetic series:
Sn = (n/2)(a1 + an),
where Sn is the sum of the series, n is the number of terms, a1 is the first term, and an is the last term.
Substituting the known values:
41 = (n/2)(41 + an)
Using the given information that the pattern is consistent, we can express an (the number of plants in the last row) in terms of the first term and the common difference. The common difference is -4 since the number of plants decreases by 4 in each row.
an = a1 + (n-1)d
an = 41 + (n-1)(-4)
an = 41 - 4n + 4
an = 45 - 4n
Substituting this into the previous equation:
41 = (n/2)(41 + 45 - 4n)
Simplifying further:
41 = (n/2)(86 - 4n)
41 = (n/2)(86) - 2n^2
82 - 4n = -2n^2 + 86
Rearranging the equation and setting it equal to 0:
2n^2 - 4n + 4 = 0
We can solve this quadratic equation using the quadratic formula:
n = (-b ± √(b^2 - 4ac))/(2a),
where a = 2, b = -4, and c = 4.
Calculating:
n = (-(-4) ± √((-4)^2 - 4(2)(4)))/(2(2))
n = (4 ± √(16 - 32))/4
n = (4 ± √(-16))/4
Since we cannot find the square root of a negative number within the real number system, it implies that there is no positive integer solution for n. This means that the number of rows will not reach 0 or a negative number, and the pattern does not continue.
Therefore, we cannot determine the exact number of plants in the last row or the total number of plants with the given information.