Question 1:Prove that 1 + 2 + 4 + ⋯ + 2^n-1 = 2^n − 1 for all natural numbers n .

Question 2:Prove that 1+1/2+1/4+⋯+1/(2^n-1)=2(1−(1/2^n) for all natural numbers n.

S = 1+ 2^{1}+2^{2}+….2^{n-1} (1)

2S = +2^{1}+2^{2}+…..+2^{n-1}+2^{n} (2) =(1)* 2

S = 2^{n} -1 (2)-(1)

S = 1+ (1/2)^{1}+(1/2)^{2}+….(1/2)^{n-1} (1)

1/2 *S = +(1/2)^{1}+(1/2)^{2}+…..+(1/2)^{n-1}+(1/2)^{n} (2) =(1)* 1/2

S*1/2 = 1 –(1/2)^{n} (1)-(2)

S = [ 1-1/2^{n}] /[1-1/2} = 2 [ 1-1/2^{n}]

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