First slide

Binomial theorem for positive integral Index

Question

If in the expansion of $(1 + x)^{n}$ , a, b, c are three consecutive coefficients, then n =

Moderate

A

$\frac{ac + ab + bc}{b^{2} + ac}$
B

$\frac{2 ac + ab + bc}{b^{2} - ac}$
C

$\frac{ab + ac}{b^{2} - ac}$
D

$\frac{2 ac + ab + bc}{b^{2} + ac}$

Solution

Here $a =^{n} C_{r - 1}, b =^{n} C_{r} and c =^{n} C_{r + 1}$
$\frac{n_{c_{r}}}{n_{c_{r - 1}}}$ $= \frac{b}{a}$ , $\Rightarrow \frac{n - r + 1}{r} = \frac{b}{a}$ $\Rightarrow an - ar + a = br$ $\Rightarrow an - ar = br - a$ $. . . . . . . . . . . . (I)$
$\Rightarrow an - (a + b) r = - a$
$\frac{n_{c_{r + 1}}}{n_{c_{r}}} = \frac{c}{b} \Rightarrow \frac{n - r}{r + 1} = \frac{c}{b} \Rightarrow b n - b r = c r + c . . . . . . . . . . . (I i)$
$s o l v e (I) a n d (I i)$
$n = \frac{2 ac + ab + bc}{b^{2} - ac}$

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