Sequences & Series
Infinite geometric series
Grade 11
Question:
<p>If \(S_p\) denotes the sum of the series \(1 + r^p + r^{2p} + \cdots\) to ∞ and \(s_p\) the sum of the series \(1 - r^p + r^{2p} - r^{3p} + \cdots\) to ∞, \(|r| < 1\), then \(S_p + s_p\) in terms of \(S_{2p}\) is</p>
<p>(1) \(2S_{2p}\)</p>
<p>(2) 0</p>
<p>(3) \(\dfrac{1}{2}S_{2p}\)</p>
<p>(4) \(-\dfrac{1}{2}S_{2p}\)</p>
Step-by-Step Solution
Key Concept: Recognize $S_p$ and $s_p$ as infinite geometric series with different common ratios ($r^p$ and $-r^p$ respectively), then apply the standard formula $\frac{a}{1-R}$ where $a$ is the first term and $R$ is the common ratio.
<p><strong>Step 1:</strong> Identify $S_p$ as a geometric series with first term $a = 1$ and common ratio $R = r^p$.</p><p>Since $|r| < 1$, we have $|r^p| < 1$, so the series converges:</p><p>$$S_p = \frac{1}{1 - r^p}$$</p><p><strong>Step 2:</strong> Identify $s_p$ as a geometric series with first term $a = 1$ and common ratio $R = -r^p$.</p><p>Since $|r| < 1$, we have $|-r^p| < 1$, so the series converges:</p><p>$$s_p = \frac{1}{1 - (-r^p)} = \frac{1}{1 + r^p}$$</p><p><strong>Step 3:</strong> The relationship between $S_p$ and $s_p$ is:</p><p>$$S_p \cdot s_p = \frac{1}{1-r^p} \cdot \frac{1}{1+r^p} = \frac{1}{(1-r^p)(1+r^p)} = \frac{1}{1-r^{2p}}$$</p><p>∴ Answer: A</p>
Correct Answer: A