I know the answer, but I can't get there: "Alicia has invented a new app..."

[aceroadholder]

I know the answer, but I can't get there: "Alicia has invented a new app..."

I have what must be a simple problem, but I'm just chasing my tail on this.
I have determined the answer by Monte Carlo method, but there must be a more elegant way to solve:
40n -2**n -40 = 0

I know that n ~ 8.1624

This must be simple, but I'm stuck.

 

[Deleted member 4993]

I have what must be a simple problem, but I'm just chasing my tail on this.
I have determined the answer by Monte Carlo method, but there must be a more elegant way to solve:
40n -2**n -40 = 0

I know that n ~ 8.1624

This must be simple, but I'm stuck.

Assuming "**" indicates exponent, the equation to solve is:

\(\displaystyle 40n- 2^n - 40 = 0\)

The first solution is 1.051828908

8.162363918 is also a solution.

I do not see a way to simple algebraic solution.

 

[aceroadholder]

Assuming "**" indicates exponent, the equation to solve is:

\(\displaystyle 40n- 2^n - 40 = 0\)

The first solution is 1.051828908

8.162363918 is also a solution.

I do not see a way to simple algebraic solution.

Thank you S.K. for your prompt reply. This question comes from the 5th sample question the New York Times gives for the NY Regents Algebra 1 Common Core exam.
www.[B]nytimes[/B].com/interactive/.../algebra-quiz.html

"Alicia has invented a new app for smart phones that two companies are interested in purchasing for a 2-year contract.
[h=3]Company A is offering her $10,000 for the first month and will increase the amount each month by $5,000.

Company B is offering $500 for the first month and will double their payment each month from the previous month.

Monthly payments are made at the end of each month. For which monthly payment will company B’s payment first exceed company A’s payment?"[/h]
The equation is derived from the second part of the word problem that is presented. "n" is the number of months in the word problem.

 

[Bob Brown MSEE]

Fixed point iteration converges rapidly in this case.

let N₀=1 then,
N_k+1 = (2^N_k + 40)/40

1

1.05

1.051763246

1.051826549

1.051828823

1.051828905

1.051828908

 

[Steven G]

Boy oh boy...this is confusing!
The 8th month is when Company B > Company A:

MONTH   COMPANY A  COMPANY B
  1      10,000        500 
  2      15,000      1,000 
  3      20,000      2,000 
  4      25,000      4,000 
  5      30,000      8,000 
  6      35,000     16,000 
  7      40,000     32,000 
  8      45,000     64,000

However, you show ~8.1624, which means "during 9th month".
That's definitely wrong.

Company A's payment, month n: 10000 + 5000(n-1) [A]
n = 8: 10000 + 5000*7 = 45000

Company B's payment, month n: 500 * 2^(n-1)
n = 8: 500 * 2^7 = 64000

When is [A] = :
10000 + 5000(n-1) = 500 * 2^(n-1)
Simplify to get:
10(n-1) - 2^(n-1) + 20 = 0

How did you get 40n -2^n - 40 = 0 ?

The solutions to the equation I show are ~7.39072 and ~-.97456
7.39072 means during 8th month...which is correct.

Anybody disagree?

Come on, somebody must be willing to disagree with Denis.

 

[Ishuda]

Thanks loads for the support, Jomo.
Join my fan club yet?
Only 1 member so far: my brother-in-law (owes me $3000).

Hey that's not true. You have at least one other member.

 

[aceroadholder]

Boy oh boy...this is confusing!
The 8th month is when Company B > Company A:

MONTH   COMPANY A  COMPANY B
  1      10,000        500 
  2      15,000      1,000 
  3      20,000      2,000 
  4      25,000      4,000 
  5      30,000      8,000 
  6      35,000     16,000 
  7      40,000     32,000 
  8      45,000     64,000

However, you show ~8.1624, which means "during 9th month".
That's definitely wrong.

Company A's payment, month n: 10000 + 5000(n-1) [A]
n = 8: 10000 + 5000*7 = 45000

Company B's payment, month n: 500 * 2^(n-1)
n = 8: 500 * 2^7 = 64000

When is [A] = :
10000 + 5000(n-1) = 500 * 2^(n-1)
Simplify to get:
10(n-1) - 2^(n-1) + 20 = 0

How did you get 40n -2^n - 40 = 0 ?

The solutions to the equation I show are ~7.39072 and ~-.97456
7.39072 means during 8th month...which is correct.

Anybody disagree?

Thanks for your analysis Denis. I see the error I made.
Where you correctly have:
When is [A] = :
10000 + 5000(n-1) = 500 * 2^(n-1)
Simplify to get:
10(n-1) - 2^(n-1) + 20 = 0

I had 10000 + 500(n-1) = 500 * 2^(n-1)
I answered the sample question correctly by quickly creating the table as you did with the simple progression. I'm an old guy and was sure that there must be a simple way to show the moment plan B's payment exceeded plan A's. Algebra class was 50+ years ago and I was a little fuzzy.