Question: $(1 + x)^{n} - nx - 1$ is divisible by (where $n \in N$)...

Question

$(1 + x)^{n} - nx - 1$ is divisible by (where $n \in N$ )

Answer 1

$(1 + x)^{n} = 1 + nx + \frac{n(n - 1)}{2!}x^{2} + \frac{n(n - 1)(n - 2)}{3!}x^{3} + .....$

⇒ $(1 + x)^{n} - nx - 1 = x^{2}\left\lbrack \frac{n(n - 1)}{2!} + \frac{n(n - 1)(n - 3)}{3!}x + ..... \right\rbrack$

From above it is clear that $(1 + x)^{n} - nx - 1$ is divisible by $x^{2}$ .

Trick : $(1 + x)^{n} - nx - 1$ . Put $n = 2$ and $x = 3$ ;

Then $4^{2} - 2.3 - 1 = 9$

Is not divisible by 6, 54 but divisible by 9. Which is given by option (3) = $x^{2} = 9$ .

Question: \((1 + x)^{n} - nx - 1\) is divisible by (where \(n \in N\))...