Find the indicated term for (1-2x)^10, middle term.
there are 11 terms, so you want the 6th.
C(10,6)*(-2x)^5
Oops, C(10,5)
To find the middle term in the expansion of (1-2x)^10, we need to determine the power of x in that term. The middle term can be found using the Binomial Theorem.
The Binomial Theorem states that for a binomial expression (a + b)^n, the expansion can be written as:
(a + b)^n = C(n, 0) * a^(n-0) * b^0 + C(n, 1) * a^(n-1) * b^1 + ... + C(n, n-1) * a^1 * b^(n-1) + C(n, n) * a^0 * b^n
where C(n, k) represents the binomial coefficient, calculated as:
C(n, k) = n! / (k! * (n-k)!)
In our case, (1-2x)^10 is the binomial expression, and we want to find the middle term, which means we need to determine the value of k.
Since the exponent of x is 10, the term with x^k will be in the middle if k = 5.
Now we can plug these values into the formula to calculate the middle term:
T(k) = C(10, 5) * 1^(10-5) * (-2x)^5
Calculating the binomial coefficient C(10, 5) using the formula C(n, k) = n! / (k! * (n-k)!) gives:
C(10, 5) = 10! / (5! * (10-5)!)
= 10! / (5! * 5!)
= (10 * 9 * 8 * 7 * 6) / (5 * 4 * 3 * 2 * 1)
= 252
Therefore, the middle term of (1-2x)^10 is given by:
T(5) = 252 * 1^(10-5) * (-2x)^5
= 252 * 1 * (-2)^5 * x^5
= 252 * (-32) * x^5
= -8,064x^5
So, the middle term in the expansion of (1-2x)^10 is -8,064x^5.