SOLUTION: 1/2+1/4+1/8+.....+1/2^n=1-1/2^n. prove by mathematical induction that above statement holds true for every integer n belongs to N. HINT:to prove that 1-1/2^(k+1)

Algebra ->  Sequences-and-series -> SOLUTION: 1/2+1/4+1/8+.....+1/2^n=1-1/2^n. prove by mathematical induction that above statement holds true for every integer n belongs to N. HINT:to prove that 1-1/2^(k+1)      Log On


   

Question 773490: 1/2+1/4+1/8+.....+1/2^n=1-1/2^n.
prove by mathematical induction that
above statement holds true for every
integer n belongs to N.

HINT:to prove that 1-1/2^(k+1)
Answer by ramkikk66(644) About Me  (Show Source):
You can put this solution on YOUR website!
To prove that:
1%2F2+%2B+1%2F4+%2B+...%2B+1%2F2%5En+=+1+-+1%2F2%5En
To prove it using induction:
1) Confirm it is true for n = 1
It is true since 1/2 = 1/2^1
2) Assume it is true for some value of n = k
i.e. 
1%2F2+%2B+1%2F4+%2B+...%2B+1%2F2%5Ek+=+1+-+1%2F2%5Ek ----> eqn (1)
3) Now prove it is true for n = k+1
i.e. the sum up to (k+1) terms = 1 - 1/2^(k+1)
Proof:
For n = k+1, the expression of the sum is:
1%2F2+%2B+1%2F4+%2B+...1%2F2%5Ek+%2B+1%2F2%5E%28k%2B1%29
=
red%281+-+1%2F2%5Ek%29+%2B+1%2F2%5E%28k%2B1%29 ---> from eqn(1)
=
%282%5E%28k%2B1%29+-+2+%2B+1%29%2F2%5E%28k%2B1%29 ---> taking common denominator 2^(k+1)
=
%282%5E%28k%2B1%29+-+1%29%2F2%5E%28k%2B1%29 ---> simplifying the numerator
=
1+-+1%2F2%5E%28k%2B1%29
Proved!
4) So we have proved that if the formula is true for n=k, it is true for 
n=k+1. Since it is true for n=1, it is proved by mathematical induction, that
it is true for all n.
Hope you got it :)