Question 773490
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To prove that:
{{{1/2 + 1/4 + ...}}}{{{+ 1/2^n = 1 - 1/2^n}}}
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. 
{{{1/2 + 1/4 + ...}}}{{{+ 1/2^k = 1 - 1/2^k}}} ----> 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:
{{{1/2 + 1/4 + ...}}}{{{1/2^k + 1/2^(k+1)}}}
=
{{{red(1 - 1/2^k) + 1/2^(k+1)}}} ---> from eqn(1)
=
{{{(2^(k+1) - 2 + 1)/2^(k+1)}}} ---> taking common denominator 2^(k+1)
=
{{{(2^(k+1) - 1)/2^(k+1)}}} ---> simplifying the numerator
=
{{{1 - 1/2^(k+1)}}}
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 :)
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