Question -
Answer -
It is known that┬а(r┬а+1)th┬аterm, (Tr+1), in the binomialexpansion of (a┬а+┬аb)n┬аis givenby┬а
.
Assuming that┬аxn┬аoccursin the (r┬а+ 1)th┬аterm of the expansion of (1+┬аx)2n, we obtain
Comparing the indices of┬аx┬аin┬аxn┬аandin┬аTr┬а+ 1, we obtain
r┬а=┬аn
Therefore, the coefficient of┬аxn┬аinthe expansion of (1 +┬аx)2n┬аis
Assuming that┬аxn┬аoccursin the (k┬а+1)th┬аterm of the expansion (1 +┬аx)2n┬атАУ1, we obtain
Comparing the indices of┬аx┬аin┬аxn┬аand┬аTk┬а+1, we obtain
k┬а=┬аn
Therefore, the coefficient of┬аxn┬аinthe expansion of (1 +┬аx)2n┬атАУ1┬аis
From (1) and (2), it is observedthat
Therefore, the coefficientof┬аxn┬аin the expansion of (1 +┬аx)2n┬аistwice the coefficient of┬аxn┬аin the expansion of (1+┬аx)2nтАУ1.
Hence, proved.