To find the total amount that Nancy saved during the 52 weeks, we need to calculate the sum of the arithmetic sequence formed by her savings.
The first term is $1, and the common difference between each subsequent term is $1 (as she saves $1 more each week).
The sum of an arithmetic sequence can be calculated using the formula: S = (n/2) * (2a + (n-1)d), where S is the sum, n is the number of terms, a is the first term, and d is the common difference.
In this case, n = 52 (52 weeks), a = 1 (first week's savings), and d = 1 (increase in savings each week).
Substituting the values into the formula, we have:
S = (52/2) * (2(1) + (52-1)(1)) = 26 * (2 + 51) = 26 * 53 = 1378
Therefore, the total amount that Nancy saved during the 52 weeks is $1,378.