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posted by  Bob on 8/17/2008 6:27:53 PM  |  status: Live  

Discrete Math-Inequalities proof by induction

Course Textbook Chapter Problem
N/A Discrete Source ISBN 0536992908 1-7 17
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Verify the inequality using induction

 (1+x)n ³ 1 + nx,  for x³ -1 and n ³ 1
 
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posted by Galois on 8/17/2008 9:00:58 PM  |  status: Live
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posted by zsm28 on 8/17/2008 9:03:07 PM  |  status: Live
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Verify the inequality using induction

 (1 + x)n ≥ 1 + nx,  for x ≠ -1 and n ≥ 1

For n = 1
(1 + x)1 = 1 + x = 1 + 1*x
so it is true for n = 1

Assume it is true for n = k, that is
 (1 + x)k ≥ 1 + kx
for n = k + 1
(1 + x)k+1
= (1 + x)*(1 + x)k
≥ (1 + x)(1 + kx)
= 1 + x + kx + kx2
= 1 + (k + 1)x + kx2
≥ 1 + (k + 1)x
 so it is also true for n = k + 1

By mathematical induction, the inequality
 (1 + x)n ≥ 1 + nx,  for x ≠ -1
is true for any positive integer n ≥ 1
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