Given positive integers i > 1, n> 2 and that the coefficients of $(3 r)^{t h}$ and $(r + 2)^{t h}$ terms in the bionomial expansion of $(1 + x)^{2 n}$ are equal, then

n=2r

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n=3r

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n=2r+1

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None of these.

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Solution

The correct option is A n=2r
$(1 + x)^{2 n}$
General term $=^{2 n} C_{r} . x^{2 n - r}$
$x^{2 n - r} = x^{3 r} x^{2 n - r} = x^{r + 2}$
$2 n - r = 3 r 2 n - r = r + 2$
$r = 3 n - 3 r 2 n = 2 r + 2$
$2 n = 4 r n = r + 1$
$n = 2 r r = n - 1$
$∴ n = 2 r$

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Given positive integers i > 1, n> 2 and that the coefficients of (3r)th and (r+2)th terms in the bionomial expansion of (1+x)2n are equal, then

The correct option is A n=2r(1+x)2nGeneral term =2nCr.x2n−rx2n−r=x3rx2n−r=xr+22n−r=3r2n−r=r+2r=3n−3r2n=2r+22n=4rn=r+1n=2rr=n−1∴ n=2r

Given positive integers i > 1, n> 2 and that the coefficients of $(3 r)^{t h}$ and $(r + 2)^{t h}$ terms in the bionomial expansion of $(1 + x)^{2 n}$ are equal, then

The correct option is A n=2r
$(1 + x)^{2 n}$
General term $=^{2 n} C_{r} . x^{2 n - r}$
$x^{2 n - r} = x^{3 r} x^{2 n - r} = x^{r + 2}$
$2 n - r = 3 r 2 n - r = r + 2$
$r = 3 n - 3 r 2 n = 2 r + 2$
$2 n = 4 r n = r + 1$
$n = 2 r r = n - 1$
$∴ n = 2 r$