The remainder , if ` 1 + 2 + 2^(2) + 2^(3) + …+ 2^(1999) ` is divided by 5, is

Options

A. 0
B. 1
C. 2
D. 3

Answer

a

Explanation

We have , ` S= (1(2^(2000) -1))/(2-1) = 2^(2000) - 1 = (2^(2))^(1000) - 1`
` = (5-1)^(1000) -1`
` = (5^(1000- 1000)C_(1) *5^(999) + ""^(1000)C_(2) *5^(998) ...+ ""^(1000)C_(998) * 5^(2) - ""^(1000)C_(999) *5 + 1) - 1`
` =5 (5^(999) - ""^(1000)C_(1) *5^(998)+ ""^(1000)C_(2) * 5^(997) -...- ""^(1000)C_(999) )`
` therefore ` Remainder is 0.