Show by induction that : ( 1 + x ) n ≥ 1 + n x, for x&gt; -1

Question

Show by induction that :  (  1  +  x      )          n        ≥  1  +  n  x, for x&amp;gt; -1

Ali Harper · Accepted Answer

This is very simple just plug -1 in for x

posader86 · Accepted Answer

Let P(n) be the statement that   (  1  +  x      )          n        1  +  n  xBasis step :P(0) is true because   (  1  +  x      )          0        =  1    (=1+0x = 1 )                                Inductive step:Assume that P(k) is true.i.e.,   (  1  +  x      )          k        1  +  k  xTo prove that P(k + 1) istrue.i.e., to show that   (  1  +  x      )          k      +      1        1  +  (  k  +  1  )  xNow   (  1  +  x      )          k      +      1        =  (  1  +  x      )          k        (  1  +  x  )(1 + kx) (1 +x)  (for n = k, we have  (  1  +  x      )          k        1  +  k  x )  =  1  +  x  +  k  x  +  k      x          2        =  1  +  (  k  +  1  )  x  +  k      x          2      1 + (k + 1)x  ∴  P  (  k  +  1  ) is true.Hence from the principleof mathematical induction   (  1  +  x      )          n        1  +  n  x is true.

Show by induction that : (1+x)^n ge 1+nx , for x > -1

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