Let the coefficients of third, fourth and fifth terms in the expansion of x + a x 2 n , x ≠ 0 , be in the ratio 12 : 8 : 3 . Then the term independent of x in the expansion, is equal to _______.

Question

Let the coefficients of third, fourth and fifth terms in the expansion of x + a x 2 n , x &#8800; 0 , be in the ratio 12 : 8 : 3 . Then the term independent of x in the expansion, is equal to _______.

Quantrex Academy · Accepted Answer

Correct answer: 0

The general term in the expansion of a + b n is T r + 1 = C r n a n - r b r . Hence, the general term in the expansion of x + a x 2 n is T r + 1 = C r n ( x ) n - r a x 2 r &#8658; T r + 1 = C r n a r x n - 3 r Given the coefficients of third, fourth and fifth terms are in the ratio 12 : 8 : 3 &#8658; C 2 n a 2 : C 3 n a 3 : C 4 n a 4 = 12 : 8 : 3 &#8658; C 2 n a 2 C 3 n a 3 = 12 8 and C 3 n a 3 C 4 n a 4 = 8 3 Using C r n C r + 1 n = r + 1 n - r , &#8658; 3 n - 2 a = 3 2 and 4 n - 3 a = 8 3 On dividing the two equ

Let the coefficients of third, fourth and fifth terms in the expansion of x + a x 2 n , x ≠ 0 , be in the ratio 12 : 8 : 3 . Then the term independent of x in the expansion, is equal to _______.

Correct answer

Step-by-step solution

More from Binomial Theorem

Let the coefficients of third, fourth and fifth terms in the expansion of x + a x 2 n , x &#8800; 0 , be in the ratio 12 : 8 : 3 . Then the term independent of x in the expansion, is equal to _______.

Correct answer

Step-by-step solution

More from Binomial Theorem

Let the coefficients of third, fourth and fifth terms in the expansion of x + a x 2 n , x ≠ 0 , be in the ratio 12 : 8 : 3 . Then the term independent of x in the expansion, is equal to _______.