Question
Fri April 02, 2010 By:

Binomial Theorem

Expert Reply
Fri April 23, 2010

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(3k) = k)

 

Now,

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.


Regards Topperlearning.

 

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