Product in thousand kilogram of a certain company in first,second and third e.t.c. Weeks is the same as the co-efficient of the first,second,third e.t.c. Power of x in the expansion of(1+x)(1-x)^-2. Find the product in the six week.
To find the product in the sixth week, we need to determine the coefficient of the sixth power of x in the expansion of (1+x)(1-x)^(-2).
First, let's expand the expression using the binomial theorem. The binomial theorem states that for any real numbers a and b, and any positive integer n, the expansion of (a + b)^n can be written as the sum of terms of the form C(n, k) * a^(n-k) * b^k.
In our case, a = 1, b = -x, and n = -2. Using the binomial theorem, we can expand the expression:
(1 + x)(1 - x)^(-2) = C(-2, 0) * 1^(-2-0) * (-x)^0 + C(-2, 1) * 1^(-2-1) * (-x)^1 + C(-2, 2) * 1^(-2-2) * (-x)^2 + ...
To find the coefficient of the sixth power of x, we need to determine the coefficient of the term with x^6 in the expansion.
Let's simplify the expansion by calculating the first few terms:
(1 + x)(1 - x)^(-2) = 1 + 3x + 6x^2 + 10x^3 + ...
We can see that the coefficient of the sixth power of x is 0. Since there is no term with x^6 in the expansion, the product in the sixth week is 0 kilograms.
Therefore, the product in the sixth week is 0 kilograms.