Have no clue how to solve this, any help?

Have no clue how to solve this, any help?

Postby Guest » Wed Dec 04, 2019 3:48 am

I can't tell if I'm just missing something obvious or if it's actually a difficult question to solve but here it is. Since I don't really care about the answer and more about how to solve this type of equation in general, showing your work on how to break it down would be appreciated. Here is he problem, rephrased with different numbers and a slightly different scenario.

Bill won the lottery and got 50,000 dollars. If went on a shopping trip each day, and spent one hundred dollars on the first day, increasing the amount he spent by 50 dollars each trip, how many does could he go shopping before he ran out of money.

Obviously I could figure this out by going 100+150+200+250+300... up to the point I hit 50,000 with the total but I'd rather not need to do that, especially when the number gets any bigger. If anyone solving this or even tell me what type of math formula this even is, I'd appreciate it.
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Re: Have no clue how to solve this, any help?

Postby Baltuilhe » Wed Dec 04, 2019 9:20 pm

Good night!

Arithmetic Progression.
[tex]a_1=100\\
r=50\\
a_n=a_1+(n-1)r\\
a_n=100+(n-1)50=100+50n-50=50+50n[/tex]

Sum:
[tex]S_n=\dfrac{\left(a_1+a_n\right)n}{2}\\
S_n\geq 50\,000\\
\dfrac{\left(a_1+a_n\right)n}{2}\geq 50\,000\\
(100+50+50n)n\geq 100\,000\\
150n+50n^2\geq 100\,000\\
50n^2+150n-100\,000\geq 0\\
n^2+3n-2\,000\geq 0[/tex]

Solving:
[tex]n<-46\;\text{or}\;n>43[/tex]

So, n=44.

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I need calculation

Postby Guest » Sun May 22, 2022 1:35 am

hy
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