It is known that the binomial coefficients of the second term, the third term and the fourth term in the expansion of (2 + x) ∧ n are arithmetic sequence! Given that the binomial coefficients of the second, third and fourth terms in the expansion of (2 + x) ∧ n are equal difference sequence! 1, find the value of N 2, and find the middle term of the expansion
The second term 2 ^ (n-1) * √ x * n -------- > the binomial coefficient is n
The third term 2 ^ (n-2) * (√ x) ^ 2 * [n * (n-1) / 2] ---- > the binomial coefficient is n * (n-1) / 2
The fourth term 2 ^ (n-3) * (√ x) ^ 3 * [n * (n-1) * (n-2) / 6] ---- > the binomial coefficient is n * (n-1) * (n-2) / 6
According to the arithmetic sequence, 2 * [n * (n-1) / 2] = n + n * (n-1) * (n-2) / 6
n*(n-1)=n+n*(n-1)*(n-2)/6
n*(n-2)=n*(n-1)*(n-2)/6
It can be seen from the question that n > 4, so n = 7
The middle terms of the expansion are 4 and 5
Item 4 2 ^ (n-3) * (√ x) ^ 3 * [n * (n-1) * (n-2) / 6
Item 5 2 ^ (n-4) * (√ x) ^ 4 * [n * (n-1) * (n-2) * (n-3) / 24]
Substituting n = 7 into computable