If the number of terms free from radicals in the expansion of 7 1 3 + 1 1 1 9 6561 is k , then the value of k 100 is equal to

Question

If the number of terms free from radicals in the expansion of 7 1 3 + 1 1 1 9 6561 is k , then the value of k 100 is equal to,

MARKS App · Accepted Answer

T r + 1 = 6561 C r 7 1 3 6561 - r 1 1 1 9 r = 6561 C r 7 6561 - r 3 1 1 r / 9 = 6561 C r 7 2187 - r 3 1 1 r 9 T r + 1 is rational if r 9 and r 3 are integers i.e. if r is a multiple of 9 . 0 ≤ r ≤ 656 1 ⇒ 0 ≤ r 9 ≤ 72 9 ⇒ r 9 = 0,1 , 2 , . . . . . , 728,72 9 ⇒ Total terms = 730 = k ⇒ k 100 = 7 . 3

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