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

\frac{a c + a b + b c}{b^{2} + a c}

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\frac{2 a c + a b + b c}{b^{2} - a c}

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\frac{a c + a c}{b^{2} - a c}

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none of these

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Solution

The correct option is B $\frac{2 a c + a b + b c}{b^{2} - a c}$
Let $a, b, c$ be three consecutive coefficients of r, (r+1) and (r+2) terms.
$\Rightarrow a =^{n} C_{r - 1}$ ....(1)
$b =^{n} C_{r}$ .....(2)
$c =^{n} C_{r + 1}$ ......(3)
Dividing (1) by (2), we get
$\frac{a}{b} = \frac{^{n} C_{r - 1}}{^{n} C_{r}}$
$\Rightarrow \frac{a}{b} = \frac{r}{n - r + 1}$
$\Rightarrow a n + a = r (a + b)$ .....(4)
Dividing (2) by (3), we get
$\frac{b}{c} = \frac{^{n} C_{r}}{^{n} C_{r + 1}}$
$\Rightarrow \frac{b}{c} = \frac{r + 1}{n - r}$
$\Rightarrow b n - c = r (b + c)$ .....(5)
Dividing (4) by (5), we get
$\frac{a n + a}{b n - c} = \frac{a + b}{b + c}$
$\Rightarrow n (b^{2} - a c) = 2 a c + b c + a b$
$\Rightarrow n = \frac{2 a c + b c + a b}{b^{2} - a c}$

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

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Δ = ∣ ∣ ∣ ∣ ∣ \begin{matrix} b^{2} - a b & b - c & b c - a c a b - a^{2} & a - b & b^{2} - a b b c - a c & c - a & a b - a^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣

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