<p>Frederick, a social media influencer, is starting a new online account at the start of a month. From the second month onwards:</p>
<ol class="bullet-list"><li><div class="list-item-content"><span class="list-item-text">His organic followers at the end of each month will be \(a\) times the total followers he had at the end of the previous month, where \(a\) is a positive constant.</span></div></li><li><div class="list-item-content"><span class="list-item-text">A company he engaged in will provide him with \(1000\) additional followers in the middle of each month.</span></div></li></ol>
<p>Let \(F(n)\) denote the total number of organic and additional followers Frederick had at the end of \(n\) months after he started his new online account, for \(n \in \mathbb{Z}^+\).</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Write down a recurrence relationship between \(F(n)\) and \(F(n+1)\) for \(n \in \mathbb{Z}^+\), giving your answer in terms of \(a\).</span><span class="inline-marks">[2]</span></div></li></ol>
<p>It is found that Frederick has \(3500\) followers at the end of the first month.</p>
<ol class="abc-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Show that \(F(3)=3500a^2+1000a+1000\).</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find an expression for \(F(n)\). Hence, find the number of months required for Frederick’s followers to exceed one million if \(a=1.5\).</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Given that the number of followers exceeds \(20000\) by the end of the first year, determine the range of values of \(a\).</span><span class="inline-marks">[2]</span></div></li></ol>
<p>Frederick finds out that, instead of the additional \(1000\) followers in the middle of every month, the company can only provide him with additional \(b\) followers in the middle of every month.</p>
<ol class="abc-list" start="5"><li><div class="list-item-content"><span class="list-item-text">Given that the number of followers remains constant since the end of the first month, find the relationship between \(a\) and \(b\).</span><span class="inline-marks">[2]</span></div></li></ol>

Frederick, a social media influencer, is starting a new online account at the start of a month. From the second month onwards:

His organic followers at the end of each month will be \(a\) times the total followers he had at the end of the previous month, where \(a\) is a positive constant.

A company he engaged in will provide him with \(1000\) additional followers in the middle of each month.

Let \(F(n)\) denote the total number of organic and additional followers Frederick had at the end of \(n\) months after he started his new online account, for \(n \in \mathbb{Z}^+\).

Write down a recurrence relationship between \(F(n)\) and \(F(n+1)\) for \(n \in \mathbb{Z}^+\), giving your answer in terms of \(a\).

It is found that Frederick has \(3500\) followers at the end of the first month.

Find an expression for \(F(n)\). Hence, find the number of months required for Frederick’s followers to exceed one million if \(a=1.5\).

Given that the number of followers exceeds \(20000\) by the end of the first year, determine the range of values of \(a\).

Frederick finds out that, instead of the additional \(1000\) followers in the middle of every month, the company can only provide him with additional \(b\) followers in the middle of every month.

Given that the number of followers remains constant since the end of the first month, find the relationship between \(a\) and \(b\).

APGP

International Baccalaureate Higher Level Mathematics

2025 HCI P1 Q13

2025 HCI P1 Q13