Find the remainder when \(2^0 + 2^1 + 2^2 + 2^3 + \dots + 2^{100}\) is divided by 7.
The remainder when 2^0 + 2^1 + 2^2 + 2^3 + ... + 2^100 is divided by 7 is 0.
This can be found using the following steps:
We can write the sum as a geometric series:
2^0 + 2^1 + 2^2 + 2^3 + ... + 2^100 = 1 + 2 + 4 + 8 + ... + 1024
The sum of a geometric series is equal to a/(1-r), where a is the first term, r is the common ratio, and n is the number of terms. In this case, a=1, r=2, and n=100.
S = a/(1-r) = 1/(1-2) = 1/(-1) = -1
The remainder when a number is divided by 7 is equal to the remainder when the number is divided by 7 and then multiplied by -1. In this case, the remainder when -1 is divided by 7 is 0.
R = (-1)/7 = 0
Therefore, the remainder when 2^0 + 2^1 + 2^2 + 2^3 + ... + 2^100 is divided by 7 is 0.
