Respuesta :
By trail and errors of options;
[tex](1.06)^{16}=2.540[/tex]
and the answer is Minimum 16 days.
What is distance?
Distance is the total movement of an object without any regard to direction.
What is Geometric Progression?
If in a sequence of terms, each succeeding term is generated by multiplying each preceding term with a constant value, then the sequence is called a geometric progression. (GP).
Explanation for the solution:
Distance covered on the first day = 2miles
Distance covered on the 2nd day = 2 + 2 х 6/100
[tex]\ \ =\ 2(1+\frac{6}{100})[/tex]
Distance covered on the 3rd day = 2(1+6/100) + 2(1+6/100) х 6/100
= [tex]2\left(1^2+2\times\left(\frac{6}{100}\right)^2\right)[/tex]
[tex]=\ 2\left(1+\frac{6}{100}\right)^2[/tex]
Distance covered on the nth day = [tex]=\ 2\left(1+\frac{6}{100}\right)^{n-1}[/tex]
Total distance covered is given by = 2 + [tex]\\ 2(1+\frac{6}{100})[/tex] [tex]+\ 2\left(1+\frac{6}{100}\right)^2[/tex]+...........
This is a Geometric Progression ,sum of G.P. is given by
[tex]S_{n} = \frac{ a(r^n - 1)}{ (r - 1)}[/tex]
Here a = 2 , [tex]r = (1+\frac{6}{100} )[/tex]
Total distance covered
[tex]=2\frac{\left(1+\frac{6}{100}\right)^{n-1}}{\left(1+\frac{6}{100}-1\right)}[/tex]
[tex]=\frac{200}{6}\left(1+\frac{6}{100}\right)^{n-1}[/tex]
[tex]=\frac{200}{6}\left(1.06\right)^{n-1}\geq51[/tex]
[tex]=33.33\left(\left({1.06}^{n-1}\right)\right)\geq51=\left({1.06}^{n-1}\right)\geq1.53015[/tex]
[tex]=1.06^{n}\geq 2.53015[/tex]
By trail and errors of options;
[tex](1.06)^{16}=2.540[/tex]
and the answer is Minimum 16 days.
Find more about distance and geometric progression:
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