2 A positive integer x is called odd if and only if there ex

2. A positive integer x is called odd if and only if there exists k Element of N such that x = 2k + 1. A positive integer x is called divisible by 3 if and only if there exists k Element of N such that x = 3k. Using these definitions, prove or disprove: For any n Element of N, if n is odd, then summation j=0 to n 2^j is divisible by 3.

Solution

2^0 + 2^1 + 2^2 ...............2^n

2^(n+1) - 1

2^(2m+2) - 1

4 * 4^m - 1 is idivsible by 3

now use mathematical inducrtion

another way of prooving it :   1 + 2 + 4 + 8 + 16

[1 + 2] + 4[1 + 2] + 16 [1 + 2]..........4^m[1 + 2]

[3] + 4[3] + 16[3] ..............4^(m)[3]

if n =2m+1 is odd then only 3 comes out coomon beacuse from every consecutive two terms 3 cpmes out coomon