The number of integral terms in the expansion of 3 1 2 + 5 1 4 680 is equal to

Step-by-step solution

Given expansion is 3 1 2 + 5 1 4 680 . The general term in the expansion x + a n is T r + 1 = C r n · x n - r · a r ⇒ T r + 1 = C r 680 3 1 2 680 - r 5 1 4 r ,   0 ≤ r ≤ 680 = C r 680 3 680 - r 2 5 r 4 For 3 680 - r 2 5 r 4 to be rational r should be a multiple of 4 . ⇒ r = 0 , 4 , 8 , 12 , . . . . . 680 ⇒ a n = a + n - 1 d ⇒ 680 = 0 + n - 1 4 ⇒ n = 171 That means r can take 171 values. Hence, the required answer is 171 .