Final answer:
To calculate the sum ∑(i + 8) for i=1 to 2ⁿ, add the sum of the first 2ⁿ natural numbers to 8 times 2ⁿ.
Step-by-step explanation:
The student is asking to determine the sum of the series given by the formula ∑(i + 8) from i = 1 to i = 2ⁿ, with n being a positive integer. We start by identifying the number of terms in the series which is equal to 2ⁿ. The sum of an arithmetic series can be found using the formula S = n/2 × (first term + last term), but this case involves a constant being added, which we need to account for separately.
To find the sum when i goes from 1 to 2ⁿ, we first sum the n terms from 1 to 2ⁿ resulting in 2ⁿ(2ⁿ + 1)/2. Then we add the constant 8 a total of 2ⁿ times which is the same as adding 8 × 2ⁿ. The final sum is S = 2ⁿ(2ⁿ + 1)/2 + 8 × 2ⁿ.