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

Lisa Hardin 2022-07-25 Answered
Show by induction that :
( 1 + x ) n 1 + n x, for x> -1
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Answers (2)

Ali Harper
Answered 2022-07-26 Author has 16 answers
This is very simple just plug -1 in for x

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posader86
Answered 2022-07-27 Author has 3 answers
Let P(n) be the statement that ( 1 + x ) n 1 + n x
Basis 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 x
To prove that P(k + 1) istrue.
i.e., to show that ( 1 + x ) k + 1 1 + ( k + 1 ) x
Now ( 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.

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