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Sagot :
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;
[tex]S_n=\frac{n}{2}[2a+(n-1)\times d[/tex]
When Sn = 150, we get;
[tex]150=\frac{n}{2} [2\times 1+(n-1)\times 6[/tex]
150 = 3·n² - 2·n
3·n² - 2·n - 150 = 0
Taking only the positive solution for n, we have;
[tex]n_{1,\:2}=\frac{-\left(-2\right)\pm \sqrt{\left(-2\right)^2-4\cdot \:3\left(-150\right)}}{2\cdot \:3}[/tex]
[tex]n_{1,\:2}=\frac{-\left(-2\right)\pm \:2\sqrt{451}}{2\cdot \:3}[/tex]
[tex]n=\frac{1+\sqrt{451}}{3},\:n=\frac{1-\sqrt{451}}{3}[/tex]
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:
https://brainly.com/question/6561461
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