Question

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

• A. $\frac{2ac+b(a+c)}{b^2-ac}$
• B. $a+b-2c$
• C. $\frac{2ac}{b^2-ac}$
• D. None of these

Answer 1

Answer: (A)

$\frac{2ac+b(a+c)}{b^2-ac}$

$a,b$ and $c$ are three consecutive coefficients of expansion, $(1+x{)}^n$
Let, $a{=}_0^n\text{C}=1$, $b{=}_1^n\text{C}=\frac{n!}{(n-1)!1!}=n$ and $c{=}_2^n\text{C}=\frac{n!}{(n-2)!2!}=n(n-1)$
Putting the value in $\frac{2ac+ab+bc}{b^2-ac}$,
$=\frac{\frac{2n(n-1)}{2}+n+\frac{n^2(n-1)}{2}}{n^2-\frac{n(n-1)}{2}}$
$=\frac{2n^2-2n+2n+n^3-n^2}{2n^2-n^2+1}$
$=\frac{n^2+n^3}{n^2+1}=n$