First slide

Introduction to P.M.I

Question

Which of the following result is valid?

Easy

A

$(1 + x)^{n} > (1 + nx)$ for all natural numbers n
B

$(1 + x)^{n} \geq (1 + nx)$ for all natural numbers z, where x > -1
C

$(1 + x)^{n} \leq (1 + nx)$ for all natural numbers n
D

$(1 + x)^{n} < (1 + nx)$ for all natural numbers n

Solution

$\begin{array}{ll} Let P (n) : (1 + x)^{n} \geq (1 + nx) \\ For n = 1, (1 + x)^{1} = 1 + x = 1 + 1 \cdot x \geq 1 + 1 \cdot x \\ (1 + x)^{1} \geq 1 + 1 \cdot x \end{array}$
For $n = k, let P (k) : (1 + x)^{k} \geq (1 + kx)$ is true.
For $n = k + 1, P (k + 1) : (1 + x)^{k + 1} \geq {1 + (k + 1) x}$ we shall show P (k + 1) is true.
consider,
$\begin{array}{l} (1 + x)^{k + 1} = (1 + x)^{k} \cdot (1 + x) \\ \geq (1 + kx) (1 + x) [if x > - 1] \\ = 1 + x + kx + {kx}^{2} \\ \geq 1 + x + kx [∵ k > 0 and x > - 1] \\ = 1 + (k + 1) x \end{array}$
Thus, $(1 + x)^{k + 1} \geq 1 + (k + 1) x, if x > - 1$

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