If n is an even positive integer greater than 1 and x 0 , then the condition that the greatest term in the expansion of 1 + x n may have the greatest coefficient also is

Question

If n is an even positive integer greater than 1 and x > 0 , then the condition that the greatest term in the expansion of 1 + x n may have the greatest coefficient also is, n - 1 n x n n - 1, n n + 1 x n + 1 n, n n + 3 x n + 3 n, n n + 2 x n + 2 n

MARKS App · Accepted Answer

If n is even, the greatest coefficient is n C n / 2 ∴ The greatest term = n C n / 2 x n / 2 ∴ n C n / 2 x n / 2 > n C n / 2 − 1 x n / 2 − 1 and n C n / 2 x n / 2 > n C n / 2 + 1 x n / 2 + 1 ⇒ n - n 2 + 1 n 2 x > 1 and n - n 2 + 1 + 1 n 2 + 1 x 1 ⇒ n / 2 n / 2 + 1 x n 2 + 1 n 2 Hence, n n + 2 x n + 2 n

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