Question

1. a company is tracking the number of complaints received on its website. during the first 4 months, they record the following numbers of complaints: 20, 25, 30, and 35. which is a possible explicit rule for the number of complaints they will receive in the nth month?

a. $a_n = 20n + 5$
b. $a_n = 15 + 5n$

Explanation:

Step1: Analyze the sequence

The given sequence of complaints is 20, 25, 30, 35. Let's check the pattern. The difference between consecutive terms is \(25 - 20 = 5\), \(30 - 25 = 5\), \(35 - 30 = 5\). So it's an arithmetic sequence with first term \(a_1 = 20\) and common difference \(d = 5\).

Step2: Recall the explicit formula for arithmetic sequence

The explicit formula for an arithmetic sequence is \(a_n=a_1+(n - 1)d\). Substituting \(a_1 = 20\) and \(d = 5\), we get \(a_n=20+(n - 1)\times5=20 + 5n-5=15 + 5n\).

We can also test the options:

• For option a: When \(n = 1\), \(a_1=20\times1 + 5=25

eq20\). So option a is incorrect.

• For option b: When \(n = 1\), \(a_1=15+5\times1 = 20\); when \(n = 2\), \(a_2=15 + 5\times2=25\); when \(n = 3\), \(a_3=15+5\times3 = 30\); when \(n = 4\), \(a_4=15+5\times4=35\). Which matches the given sequence.

Answer:

b. \(a_{n}=15 + 5n\)