If the sum of the coefficients in the expansions of (1 + 2x)^n is 6561, the greatest term in the expansion for x=1/2 is
The sum of coefficients in the expansion of any general (1 + ax)n can be obtained by putting x = 1, as it will make all the appearences of x in the expansion as 1 and the expansion will then consists of the sum of all the coefficients.
put x = 1 in (1 + 2x)n , we get,
(1 + 2)n = 6561
=> 3n = 6561
Taking log with base 3 on both sides,
=> n*log3(3) = log3(6561)
=> n = log3(38) ( as, log3(3) = 1)
=> n = 8 ( as, log3(3k) = k)
when x = 1/2,
the given expression becomes = (1 + 2* 1/2 )8
= (1 + 1)8
The general rth term of this expansion is = 8Cr (1)r = 8Cr
Therefore, the greatest term of this expansion will the one with maximum value of 8Cr.
We know that when n = even integer, then the maximum value of nCr is attained at r = n/2,
so, the maximum value of 8Cr will be attained at r = 4.
=> 8C4 = 8! / 4! (8-4)! = 8! / 4!*4! = 70.