If A and B are the coefficient of xn in the expansion of 1-x2 n and 1-x2 n-1 respectively- then AB -

The coefficient of xn is (1 + x)2n is ? Since Tr +1 = 2nCr xr For xn coefficient put r = n i.e coefficient of xn is 2nCn nbsp; i.e. A = 2nCn and for coeffici ...

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Byju's Answer

Standard VIII

Mathematics

Multiplication of Any Polynomial

If A and B ar...

Question

If A and B are the coefficient of xⁿ in the expansion of (1 + x)²ⁿ and (1 + x)^{2n – 1} respectively, then $\frac{A}{B} =$ ___________.

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Solution

The coefficient of xⁿ is (1 + x)²ⁿ is ?
Since T_{r +1} = ²ⁿC_r x^r
For xⁿcoefficient put r = n
i.e coefficient of xⁿis ²ⁿC_n
i.e. A = ²ⁿC_n
and for coefficient of xⁿin (1 + x)^2n–1
T_{r +1} = ^2n–1C_r x^r
Put r = n
i.e. coefficient of xⁿin (1 + x)^2n–1 is ^2n–1C_n
i.e B = ^2n–1C_n
$\begin{array}{rcl} ∴ \frac{A}{B} & = & \frac{^{2 n} C_{n}}{^{2 n - 1} C_{n}} = \frac{(2 n)! n! n!}{n! n! (2 n - 1)!} \\ = & 2 \end{array}$

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If A and B are the coefficient of xn in the expansion of (1 + x)2n and (1 + x)2n – 1 respectively, then AB=___________.

If A and B are the coefficient of xⁿ in the expansion of (1 + x)²ⁿ and (1 + x)^{2n – 1} respectively, then $\frac{A}{B} =$ ___________.