Question

recall that $a_n = a_1 + d(n - 1)$ and recall that $a_1 =$ some value and $a_n = a_{n - 1} + d$

if a person starts a walking program, they start with.5 miles each day for the first week. each week they add 0.25 miles to their daily walk (so on week 2, they would walk 1.25 miles each day). if this program continues, how many miles will they walk each day during the 20th week?

\bigcirc 14 miles

\bigcirc 5.25 miles

\bigcirc 5 miles

\bigcirc 4.25 miles

\bigcirc 4 miles

Explanation:

Step1: Identify arithmetic sequence parameters

The problem is about an arithmetic sequence. Here, \(a_1 = 0.5\) (miles in the first week), \(d = 0.25\) (common difference, miles added each week), and we need to find \(a_{20}\) (miles in the 20th week). The formula for the \(n\)-th term of an arithmetic sequence is \(a_n=a_1 + d(n - 1)\).

Step2: Substitute values into the formula

Substitute \(a_1 = 0.5\), \(d = 0.25\), and \(n = 20\) into the formula:

$$
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Wait, there is a mistake in the previous thought. Wait, let's recalculate: \(20 - 1=19\), \(0.25\times19 = 4.75\), \(0.5+4.75 = 5.25\)? But wait the option has 5.25. Wait, let's check the problem again. Wait, the first week: 0.5 miles. Week 2: 0.5 + 0.25=0.75? Wait no, the problem says "so on week 2, they would walk 1.25 miles each day". Wait, that means my initial \(a_1\) is wrong. Wait, the problem says "start with 0.5 miles each day for the first week. Each week they add 0.25 miles". Wait, week 1: 0.5, week 2: 0.5+0.25 = 0.75? But the problem says week 2 is 1.25. Oh! Wait, maybe the first term is 1? Wait no, the problem says "start with 0.5 miles each day for the first week. Each week they add 0.25 miles to their daily walk (so on week 2, they would walk 1.25 miles each day)". Wait, that means my understanding is wrong. Wait, 0.5 + 0.75? No, maybe the first term \(a_1 = 1\)? Wait no, the problem says "start with 0.5 miles each day for the first week". Wait, maybe the problem has a typo, but according to the formula given \(a_n=a_1 + d(n - 1)\), and the example says week 2 is 1.25. Let's use the example to find \(a_1\) and \(d\). For \(n = 2\), \(a_2=1.25\). If \(a_n=a_1 + d(n - 1)\), then \(1.25=a_1 + d(2 - 1)=a_1 + d\). If the first week is \(n = 1\), \(a_1 = 0.5\), then \(d=1.25 - 0.5 = 0.75\)? But the problem says "each week they add 0.25 miles". There is a contradiction. Wait, maybe the problem's "start with 0.5 miles" is wrong, or the example is wrong. Wait, let's re - examine the problem: "they start with.5 miles each day for the first week. Each week they add 0.25 miles to their daily walk (so on week 2, they would walk 1.25 miles each day)". Wait, 0.5+0.75 = 1.25, so \(d = 0.75\)? But the problem says "add 0.25 miles". This is a problem. But according to the formula given \(a_n=a_1 + d(n - 1)\), and if we take \(a_1 = 0.5\), \(d = 0.25\), \(n = 20\): \(a_{20}=0.5+0.25\times19=0.5 + 4.75 = 5.25\). So the correct answer should be 5.25 miles.

Answer:

5.25 miles (the option with 5.25 miles)