(1 + x)^n - nx - 1 divisible (where n N) - Vidyadip

MCQ

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

A
by $2x$
✓
by ${x^2}$
C
by $2{x^3}$
D
All of these

✓

Answer

Correct option: B.

by ${x^2}$

b
(b) ${(1 + x)^n} = 1 + nx + \frac{{n(n - 1)}}{{1.2}}{x^2} + \frac{{n(n - 1)(n - 2)}}{{1.2.3}}{x^3} + ....$

${(1 + x)^n} - nx - 1 = {x^2}\left[ {\frac{{n(n - 1)}}{{1.2}} + \frac{{n(n - 1)(n - 2)}}{{1.2.3}}x + ....} \right]$

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

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