Statistics
Arithmetic Mean
Grade None
Question:
<p>The AM of the series \(1, 2, 4, 8, 16, \ldots, 2^n\) is</p>
<p>\(\dfrac{2^n - 1}{n}\)</p>
<p>\(\dfrac{2^{n+1} - 1}{n+1}\)</p>
<p>\(\dfrac{2^n + 1}{n}\)</p>
<p>\(\dfrac{2^n - 1}{n+1}\)</p>
Step-by-Step Solution
Key Concept: The arithmetic mean equals the sum of all terms divided by their count. For a geometric series with first term a, common ratio r, and (n+1) terms, use the formula for geometric series sum: S = a(r^(n+1) - 1)/(r - 1).
<p><strong>Step 1:</strong> Identify the series structure. The series is 1, 2, 4, 8, ..., 2^n, which is a geometric series with first term a = 1, common ratio r = 2, and (n+1) total terms.</p><p><strong>Step 2:</strong> Calculate the sum using the geometric series formula: S = a(r^(n+1) - 1)/(r - 1) = 1·(2^(n+1) - 1)/(2 - 1) = 2^(n+1) - 1</p><p><strong>Step 3:</strong> Apply the arithmetic mean formula: AM = (Sum of terms)/(Number of terms) = (2^(n+1) - 1)/(n + 1)</p><p>∴ Answer: D (which should be $\frac{2^{n+1} - 1}{n+1}$)</p>
Correct Answer: D