First slide

Introduction to P.M.I

Question

Let $x > - 1$ , then statement $P (n) : (1 + x)^{n} > 1 + n x$ is true for

Moderate

A

all $n \in N$
B

all $n > 1$
C

all $n > 1$ provided $x \neq 0$
D

none of these

Solution

For n=2,
$P (2) : (1 + x)^{2} = 1 + 2 x + x^{2} > 1 + 2 x$ ,as $x \neq 0$ .
Assume that
$P (k) : (1 + x)^{k} > 1 + k x$ (1)
for some $k \in N, k > 1$
As $x > - 1$ , multiplying both the sides of (1) by $1 + x$ , we get
$\begin{array}{l} (1 + x)^{k + 1} > (1 + k x) (1 + x) \\ = 1 + (k + 1) x + k x^{2} \\ > 1 + (k + 1) x \end{array}$
$[∵ k x^{2} > 0]$
Thus, $P (k + 1)$ is true.
By the principle of mathematical induction $P (n)$ is true for
all $n > 1$ provided $x \neq 0$ .

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