The arrangement of the rose plants on the triangular plot is such that they
form a series or progression that is defined.
The number of rows on Bill's plot is; 8 rows
The given parameters for the triangular plot are;
Number of plants at the corner = 1 plant
Number of additional plants per row = 6 plants
Number of rose plants = 150 rose plants
The number of rows in the plot.
The difference between successive rows, d = 6
The number rose at the top vertex, a = 1
Therefore, the rose in the garden forms an arithmetic progression
The first term, a = 1
The common difference, d = 6
The number of rows Bill's plot will have, n is given by the sum of n in terms of
an arithmetic progression, Sn, is given as follows;
![S_n=\frac{n}{2}[2a+(n-1)\times d](https://tex.z-dn.net/?f=S_n%3D%5Cfrac%7Bn%7D%7B2%7D%5B2a%2B%28n-1%29%5Ctimes%20d)
When Sn = 150, we get;
![150=\frac{n}{2} [2\times 1+(n-1)\times 6](https://tex.z-dn.net/?f=150%3D%5Cfrac%7Bn%7D%7B2%7D%20%5B2%5Ctimes%201%2B%28n-1%29%5Ctimes%206)
150 = 3·n² - 2·n
3·n² - 2·n - 150 = 0
Taking only the positive solution for n, we have;
![n_{1,\:2}=\frac{-\left(-2\right)\pm \sqrt{\left(-2\right)^2-4\cdot \:3\left(-150\right)}}{2\cdot \:3}](https://tex.z-dn.net/?f=n_%7B1%2C%5C%3A2%7D%3D%5Cfrac%7B-%5Cleft%28-2%5Cright%29%5Cpm%20%5Csqrt%7B%5Cleft%28-2%5Cright%29%5E2-4%5Ccdot%20%5C%3A3%5Cleft%28-150%5Cright%29%7D%7D%7B2%5Ccdot%20%5C%3A3%7D)
![n_{1,\:2}=\frac{-\left(-2\right)\pm \:2\sqrt{451}}{2\cdot \:3}](https://tex.z-dn.net/?f=n_%7B1%2C%5C%3A2%7D%3D%5Cfrac%7B-%5Cleft%28-2%5Cright%29%5Cpm%20%5C%3A2%5Csqrt%7B451%7D%7D%7B2%5Ccdot%20%5C%3A3%7D)
![n=\frac{1+\sqrt{451}}{3},\:n=\frac{1-\sqrt{451}}{3}](https://tex.z-dn.net/?f=n%3D%5Cfrac%7B1%2B%5Csqrt%7B451%7D%7D%7B3%7D%2C%5C%3An%3D%5Cfrac%7B1-%5Csqrt%7B451%7D%7D%7B3%7D)
The number of rows Bill's plot has, n ≈ 7.3965
Given that the 7th row is completed, an 8th row will be present on Bill's plot
The number of rows Bill's plot will have = 8 rows
To learn more about the arithmetic progression visit:
brainly.com/question/6561461
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