Question

The arithmetic mean of the series \(1, 2, 4, 8, 16, \dots, 2^n\) is

• A. \(\frac{2^n - 1}{n}\)
• B. \(\frac{2^{n+1} - 1}{n+1}\)
• C. \(\frac{n+1}{2^n + 1}\)
• D. \(\frac{2^n - 1}{n+1}\)

Hint:

Don't forget that from $2^0$ to $2^n$ there are $(n+1)$ terms! Sum $= 2^{n+1}-1$, Mean $= \frac{2^{n+1}-1}{n+1}$.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The arithmetic mean of a sequence is defined as the sum of all elements divided by the total number of elements.
Key Formula or Approach:
\[ \text{Arithmetic Mean } (\bar{x}) = \frac{S_N}{N} \]
\[ S_N = \frac{a(r^N - 1)}{r - 1} \quad \text{(Sum of GP)} \]

Step 2: Detailed Explanation:
The given series is:
\[ 2^0, 2^1, 2^2, 2^3, \dots, 2^n \]
- First term: \(a = 1\)
- Common ratio: \(r = 2\)
- Total number of terms: \(N = n + 1\) (from power 0 to power \(n\))

Step 1: Calculate the sum of the geometric progression:
\[ S_{n+1} = \frac{a(r^{n+1} - 1)}{r - 1} = \frac{1(2^{n+1} - 1)}{2 - 1} = 2^{n+1} - 1 \]
Compute the arithmetic mean:
\[ \bar{x} = \frac{S_{n+1}}{N} = \frac{2^{n+1} - 1}{n + 1} \]

Step 3: Final Answer:
Thus, the arithmetic mean is \(\frac{2^{n+1} - 1}{n+1}\), matching option (B).