Question

a donut company has 2 stores in the first year of business. the company is considering two plans for expanding its chain of stores. a table for plan a and plan b is shown below. at the start of which year will plan b have more stores? a year 5 b year 6 c year 7 d year 8

Explanation:

Step1: Analyze Plan A's pattern

Looking at Plan A: Year 1 (2), Year 2 (17), Year 3 (32), Year 4 (47), Year 5 (62). The difference between consecutive years: \(17 - 2 = 15\), \(32 - 17 = 15\), \(47 - 32 = 15\), \(62 - 47 = 15\). So Plan A is an arithmetic sequence with common difference \(d = 15\). The formula for the \(n\)-th term of an arithmetic sequence is \(a_n=a_1+(n - 1)d\), where \(a_1 = 2\), \(d = 15\). So \(a_n=2+(n - 1)\times15=15n - 13\).

Step2: Analyze Plan B's pattern

Looking at Plan B: Year 1 (2), Year 2 (4), Year 3 (8), Year 4 (16), Year 5 (32). This is a geometric sequence with first term \(b_1 = 2\) and common ratio \(r = 2\). The formula for the \(n\)-th term of a geometric sequence is \(b_n=b_1\times r^{n - 1}\), so \(b_n=2\times2^{n - 1}=2^n\).

Step3: Calculate values for Year 6,7,8

• Year 6 (\(n = 6\)):
• Plan A: \(a_6=15\times6 - 13=90 - 13 = 77\)
• Plan B: \(b_6=2^6 = 64\). Here, \(77>64\) (Plan A has more)
• Year 7 (\(n = 7\)):
• Plan A: \(a_7=15\times7 - 13=105 - 13 = 92\)
• Plan B: \(b_7=2^7 = 128\). Here, \(128>92\) (Plan B has more)

Answer:

C. Year 7