Question

a student exercises 5 minutes the first day, 12 minutes the second day, and 19 minutes the third day. this situation models an arithmetic sequence. on day 15, how many minutes does the student exercise?
a 98 minutes
b 110 minutes
c 103 minutes
d 75 minutes

Explanation:

Step1: Identify the arithmetic sequence

The first term \(a_1 = 5\), common difference \(d = 12 - 5 = 7\) (or \(19 - 12 = 7\)).

Step2: Use the arithmetic sequence formula

The formula for the \(n\)-th term of an arithmetic sequence is \(a_n = a_1 + (n - 1)d\). For \(n = 15\), substitute \(a_1 = 5\), \(d = 7\), and \(n = 15\) into the formula:
\(a_{15} = 5 + (15 - 1) \times 7\)

Step3: Calculate the value

First, calculate \((15 - 1) \times 7 = 14 \times 7 = 98\). Then, \(a_{15} = 5 + 98 = 103\). Wait, no, wait, the first term is 5, second 12 (5+7), third 19 (12+7), so the formula is correct. Wait, but let's recalculate: \(a_n = a_1 + (n - 1)d\), so \(a_{15} = 5 + (15 - 1)7 = 5 + 147 = 5 + 98 = 103\)? Wait, but the options have C as 103 minutes. Wait, but let me check again. Wait, the first day is 5, second 12 (5+7), third 19 (12+7), so the common difference is 7. So for day 15, it's 5 + (15 - 1)7 = 5 + 98 = 103. So the answer should be C. 103 minutes. Wait, but maybe I misread the problem. Wait, the problem says "how many minutes does the student exercise on day 15". So yes, arithmetic sequence, first term 5, common difference 7. So the 15th term is 5 + 147 = 5 + 98 = 103. So the correct option is C. 103 minutes.

Answer:

C. 103 minutes