Question

Найдите средние члены биномов (x +a ^8) и( x +b ^9) Почему у них число средних членов не равно?

Answer 1

(х+а)^8=a^8+8*a^7*x+28*a^6*x^2+56*a^5*x^3+70*a^4*x^4+56*a^3*x^5+28*a^2*x^6+8*a*x^7+x^8
(x+b)^9=b^9+9*b^8*x+36*b^7*x^2+84*b^6*x^3+126*b^5*x^4+126*b^4*x^5+84*b^3*x^6+36*b^2*x^7+9*b*x^8+x^69
Количество средних членов бинома зависит от степени бинома, чем больше степень, тем больше средних членов бинома.

Answer 2

Calculating the number of terms in (x + a)^8 and (x + b)^9 binomial expansions

The number of terms in a binomial expansion is determined by the binomial coefficient formula:

C(n, k) = n! / (k!(n - k)!)

Where: - n is the exponent of the binomial - k is the specific term number we want to calculate - ! denotes the factorial operation

For (x + a)^8: - n = 8 - To find the total number of terms, we set k = 0, 1, 2, ..., n - Using the binomial coefficient formula, the number of terms is C(8, 0) + C(8, 1) + ... + C(8, 8) = 56

For (x + b)^9: - n = 9 - Using the same process, the number of terms is C(9, 0) + C(9, 1) + ... + C(9, 9) = 126

Therefore, the number of terms in (x + a)^8 and (x + b)^9 are not equal because the exponents n are different (8 vs 9). The exponent determines the maximum value of k, and thus the total number of terms when summed using the binomial coefficient formula.