The greatest term in the expansion of (1+x)¹⁵ , when x= 1 2 is

Step-by-step solution

Let T_ r+1 and T_r denotes the ( r +1) th and r th term in the expansion of (1+x)¹⁵ T_ r+1 = ^n C_r x^r and T_r= ^n C_ r-1 x^ r-1 , where n=15 For greatest term in the expansion aligned & T_ r+1 T_r ^n C_r x^r ^n C_ r-1 x^ r-1 & n ! x^r r^ !(n-r) ! n ! x^ r-1 (r-1) !(n-r+1) ! aligned aligned & x r (r-1) !( n - r ) ! 1 (r-1) ! (n-r+1) (n-r) ! & x r 1 (n-r+1) & 1 r ( 1 2 ) 1 (16-r) ( n=15 and x= 1 2 ) & 16-r 2 r & 3 r 16 r 16 3 aligned Hence r=5 will be the greatest term T_ r+1 =T_ 5+1 = ¹⁵ C₅ x^5= ¹⁵ C₅ ( 1 2 )^5= 1