First slide

Binomial theorem for positive integral Index

Question

If for positive integers $r > 1, n > 2$ , the coefficient of the (3r)th and (r + 2)th powers of X in the expansion of (1 + x)²ⁿ are equal, then

Moderate

A

$n = 2 r$
B

$n = 3 r$
C

$n = 2 r + 1$
D

None of these

Solution

In the expansion of $(1 + x)^{2 n},$ the general term
As given for
$=^{2 n} C_{k} x^{k}, 0 \leq k \leq 2 n r > 1, n > 2,^{2 n} C_{3 r} =^{2 n} C_{r + 2}$
$\Rightarrow Either 3 r = r + 2 or 3 r = 2 n - (r + 2) [∵^{n} C_{r} =^{n} C_{n - r}]$
$r = 1 or n = 2 r + 1$
We take the relation onIy
$n = 2 r + 1 [∵ r > 1]$

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