If for some positive integer  nthe coefficients of three consecutive  terms in the binomial  expansion of (1+x)n+5are in the ratio  5: 10; 14,. then the largest coefficient- in the expansion is

Let three consecutive  terms be Tr,Tr+1 and Tr+2 Their coefficients are  n+5Cr−1,n+5Cr  and  n+5Cr+1 .It is given that  n+5Cr−1:n+5Cr:n+5Cr+1=5:10:14 n+5Cr n+5Cr−1=105 and  n+5Cr+1 n+5Cr=1410⇒n+5−r+1r=2  and n+5−rr+1=75 ∵nrnr−1=n−r+1r⇒n−r+6=2r and  5(n−r+5)=7r+7⇒n−3r+6=0 and 5n−12r+18=0⇒n=6,r=4 Thus, the coefficients in the binomial expansion of (1+x)11 are  11C0,11C1,…,11C11.The greatest coefficient  is  11C5=11C6=462