Question

alicia has invented a new app that two companies are interested in purchasing for a 2-year contract. - company a is offering $10,000 for the first month and will increase the amount each following month by $5000. - company b is offering $500 for the first month and will double their payment each following month. for which monthly payment will company bs payment first exceed company as payment?

Explanation:

Step1: Define Company A's payment formula

Company A's payment follows an arithmetic sequence. The formula for the payment in month $n$ is:
$A(n) = 10000 + 5000(n-1)$
Simplify: $A(n) = 5000n + 5000$

Step2: Define Company B's payment formula

Company B's payment follows a geometric sequence. The formula for the payment in month $n$ is:
$B(n) = 500 \times 2^{n-1}$

Step3: Solve $B(n) > A(n)$

Set up the inequality:
$500 \times 2^{n-1} > 5000n + 5000$
Divide both sides by 500:
$2^{n-1} > 10n + 10$

Step4: Test integer values of $n$

• $n=6$: $2^{5}=32$; $10(6)+10=70$ → $32 < 70$
• $n=7$: $2^{6}=64$; $10(7)+10=80$ → $64 < 80$
• $n=8$: $2^{7}=128$; $10(8)+10=90$ → $128 > 90$

Answer:

8