Binomial Theorem~20點

1.)
The coefficient of the 2nd term in the expansion of (3 - x)^n in descending powersof x is 15, where n is a positive integer. Find the value of n.

Solution :
(a + b)^n = a^n + nC1*a^(n-1)*b + nC2*a^(n-2)*b^2+ ...... + nCn-1*a*b^(n-1) + b^n

the 2nd term in the expansion of (3 - x)^n in descending powers of x :
nCn-1 ´ (3) ´ (-x)^14 = 15x^14
[n!/(n - 1)!] ´ 3 = 15
3n = 15
n = 5


2.)
The coefficient of the 3nd term in the expansion of (x-1/3x)^n indescending powers of x is 28/9,where n is a positive integer. Find the value ofn.

Solution :
(a + b)^n = a^n + nC1*a^(n-1)*b + nC2*a^(n-2)*b^2+ ...... + nCn-1*a*b^(n-1) + b^n

The 3rd term in the expansion of [x - (1/3x)]^n in descending powers of x :
nC2 ´ (1/3)^2 = 28/9
[n!/2!(n - 2)!] ´ (1/9) = 28/9
n(n - 1)/2 = 28
n² - n = 56
n² - n - 56 = 0
(n + 7)(n - 8) = 0
n = -7 (rejected) or n = 8