Let the coefficients of three consecutive terms in the binomial expansion of ( 1 + 2 x ) n be in the ratio 2 :   5 :   8 . Then the coefficient of the term, which is in the middle of these three terms, is

Step-by-step solution

Given, The coefficients of three consecutive terms in the binomial expansion of ( 1 + 2 x ) n be in the ratio 2 :   5 :   8 . Now r t h term in expansion of ( 1 + 2 x ) n is given by, t r + 1 = C r n ( 2 x ) r Now let T r ,   T r + 1 ,   T r + 2 are in the ratio 2 : 5 : 8 ⇒ T r T r + 1 = C r - 1 n ( 2 ) r - 1 C r n ( 2 ) r = 2 5 ⇒ n ! ( r - 1 ) ! ( n - r + 1 ) ! n ! ( 2 ) r ! ( n - r ) ! = 2 5 ⇒ r n - r + 1 = 4 5 ⇒ 5 r = 4 n - 4 r + 4 ⇒ 9 r = 4 ( n + 1 )   . .