<p>The school's step challenge is an initiative by the PE department to encourage students to lead more active lifestyles. The table below shows the corresponding daily step counts and the number of health points that can be earned.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="width: 100%;"><thead><tr><th colspan="2" style="text-align: center; --cell-justify: center"><p style="text-align: center;"><strong>Steps Category</strong></p></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center"><p><strong>Steps Counts (Daily)</strong></p></td><td style="text-align: center; --cell-justify: center"><p><strong>Health Points (Daily)</strong></p></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>$5000 - 7499 $</p></td><td style="text-align: center; --cell-justify: center"><p>$5$</p></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>$7500 - 9999 $</p></td><td style="text-align: center; --cell-justify: center"><p>$15$</p></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>$10000$ and more</p></td><td style="text-align: center; --cell-justify: center"><p>$25$ (Max)</p></td></tr></tbody></table></div></div>
<p>Student $A$ walks $4500$ steps on the first day. For each subsequent day, he increases his step count by $350$ steps from the previous day.</p>
<ol class="abc-list"><li><ol class="i-list"><li><div class="list-item-paragraph">On which day will Student $A$ first earn $15$ Health Points in one day?<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">How many Health Points would Student $A$ earn by the end of the $20$th day?<span class="inline-marks">[3]</span></div></li></ol></li></ol>
<p>Student $B$ walks $7500$ steps on the first day. For each subsequent day, he increases his step count by $x \%$ from the previous day.</p>
<ol class="abc-list" start="3"><li><ol class="i-list"><li><div class="list-item-paragraph">Student $B$ would like to complete a total of at least $261,500$ steps in the first $20$ days. Find the minimum value of $x$, giving your answer correct to $2$ decimal points.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Given that&nbsp;$x = 3.80$, find the total number of health points that Student $B$ can earn in $20$ days.<span class="inline-marks">[3]</span></div></li></ol></li></ol>
<ol class="abc-list" start="4"><li><div class="list-item-paragraph">A student earns a $\$2$ coupon for completing at least $10,000$ steps on Day $1$. For each subsequent day, the student will earn a new coupon and the value of each new coupon decreases by $9 \%$ from the value of the previous day's coupon. If the student consistently completes $10,000$ steps every day, show that the total coupon value accumulated over time will not exceed $\$23$.<span class="inline-marks">[2]</span></div></li></ol>

The school's step challenge is an initiative by the PE department to encourage students to lead more active lifestyles. The table below shows the corresponding daily step counts and the number of health points that can be earned.

<p style="text-align: center;"><strong>Steps Category</strong></p>

<p><strong>Steps Counts (Daily)</strong></p>

<p><strong>Health Points (Daily)</strong></p>

Student $A$ walks $4500$ steps on the first day. For each subsequent day, he increases his step count by $350$ steps from the previous day.

Student $B$ walks $7500$ steps on the first day. For each subsequent day, he increases his step count by $x \%$ from the previous day.

A student earns a $\$2$ coupon for completing at least $10,000$ steps on Day $1$. For each subsequent day, the student will earn a new coupon and the value of each new coupon decreases by $9 \%$ from the value of the previous day's coupon. If the student consistently completes $10,000$ steps every day, show that the total coupon value accumulated over time will not exceed $\$23$.