Question

If (a + x) n is expanded in increasing order of x, find (i) the value of n if the coefficients of x^2 and x^n-2 are equal

Answer 1

The value of n of the coefficients of x² and xⁿ⁻² are equal is 4

Step-by-step explanation:

$(a+x)^n=a^n(1+\frac{x}{a})^n$

$=a^n[^nC_0+^nC_1(\frac{x}{a})+^nC_2(\frac{x}{a})^2+^nC_3(\frac{x}{a})^3+......+^nC_{n-2}(\frac{x}{a})^{n-2}+^nC_{n-1}(\frac{x}{a})^{n-1}+^nC_n(\frac{x}{a})^n]$

Coefficient of $x^2$ and $x^{n-2}$ are equal

Therefore,

$a^n(\frac{^nC_2}{a^2})=a^n(\frac{^nC_{n-2}}{a^{n-2}})$

$\implies \frac{1}{a^2}\times\frac{n!}{2!(n-2)!}=\frac{1}{a^{n-2}}\times\frac{n!}{2!(n-2)!}$

$\implies a^{n-2}=a^2$

$\implies \frac{a^n}{a^2}=a^2$

$\implies a^n=a^4$

$\implies n=4$        (Comparing the powers on both sides)

Hope this answer is helpful.

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