The mean of the series 1, 2, 4, 8...., 2ⁿ is given as: 

Key Points 

• To find the mean of the series 1, 2, 4, 8, ..., 2n, we need to sum up all the terms in the series and divide by the total number of terms.
• The given series follows a pattern where each term is obtained by multiplying the previous term by 2. So, we can express the terms in the series as 2⁰, 2¹, 2², 2³, ..., 2^(n-1), 2ⁿ.
• The sum of the terms can be calculated as: 2⁰ + 2¹ + 2² + 2³ + ... + 2^(n-1) + 2ⁿ
• This is a geometric series with a common ratio of 2. The sum of a geometric series can be calculated using the formula: Sum = \(a \times (r^n - 1) \over (r - 1)\), where "a" is the first term, "r" is the common ratio, and "n" is the total number of terms.
• In this case, the first term "a" is 2⁰ = 1, the common ratio "r" is 2, and the total number of terms "n" is n+1.
• Substituting these values into the formula, we have: Sum = \(1 \times (2^{(n+1)} - 1 \over (2 - 1)\).
  ⇒ sum = \(= {2^{(n+1)} - 1 \over 1} = 2^{(n+1)} - 1\)
• To find the mean, we divide the sum by the total number of terms, which is n+1: Mean = \(2^{(n+1)} - 1 \over (n + 1)\)

Hence, the mean of the series 1, 2, 4, 8, ..., 2n is given by the expression \(2^{(n+1)} - 1 \over (n + 1)\).