Final answer:
The sum of all terms in the given geometric progression, which starts with 1 and ends with 64 with a common ratio of 2, is 127.
Step-by-step explanation:
The question requires us to find the sum of all terms in a geometric progression with the first term (a) as 1, the last term (l) as 64, and the common ratio (r) as 2. To find the sum of this geometric series, we can use the formula for the sum of a finite geometric series: S = a(1 - r^n) / (1 - r), where n is the number of terms in the series.
The number of terms (n) can be found from the relationship l = ar^(n-1), which can be written, in this case, as 64 = 1*2^(n-1), leading to n = 7 after solving the equation. Now, knowing the number of terms and the values for a and r, we can calculate the sum of the progression.
Sum, S = 1(1 - 2^7) / (1 - 2) = (1 - 128) / (-1) = 127. Thus, the sum of all terms of the progression is 127.