Introduction to Math Proof

Mathematics
Tutor: None Selected Time limit: 1 Day

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.

Feb 22nd, 2015

S   = 1+ 21+22+….2n-1                  (1)

2S   =  +21+22+…..+2n-1+2n                  (2)  =(1)* 2

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

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Feb 22nd, 2015

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