CBSE Class 12-science Answered

Dear Student,

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.

So,

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(3^k) = k)

Now,

when x = 1/2,

the given expression becomes = (1 + 2* 1/2 )⁸

= (1 + 1)8

The general r^th term of this expansion is = ⁸C_r (1)^r = ⁸C_r

Therefore, the greatest term of this expansion will the one with maximum value of ⁸C_r.

We know that when n = even integer, then the maximum value of ⁿC_r is attained at r = n/2,

so, the maximum value of ⁸C_r will be attained at r = 4_.

=> ⁸C₄ = 8! / 4! (8-4)! = 8! / 4!*4! = 70.

Regards Topperlearning.