Find the first the 3rd term in the binomial expansion of (2-3x)^10 in ascending powers of x
that would be C(10,2) * 2^(10-2) (-3x)^2 = 103680x^2
To find the first and third terms in the binomial expansion of (2-3x)^10 in ascending powers of x, we can use the binomial theorem.
The binomial theorem states that for any positive integer n, the nth power of a binomial (a + b)^n can be expanded as:
(a + b)^n = C(n, 0)a^n + C(n, 1)a^(n-1)b + C(n, 2)a^(n-2)b^2 + ... + C(n, n-1)ab^(n-1) + C(n, n)b^n
Where C(n, k) represents the binomial coefficient, defined as C(n, k) = n! / (k!(n-k)!).
In this case, we have (2-3x)^10. So, let's find the first term:
The first term will be when k = 0 in C(10, k)a^(10-k)b^k.
So, the first term is C(10, 0)(2^10)(-3x)^0 = (1)(2^10)(1) = 2^10 = 1024.
Now, let's find the third term:
The third term will be when k = 2 in C(10, k)a^(10-k)b^k.
So, the third term is C(10, 2)(2^8)(-3x)^2.
C(10, 2) = 10! / (2!(10-2)!) = 10! / (2!8!) = (10*9) / (2*1) = 45.
Therefore, the third term is 45 * (2^8) * (-3x)^2 = 45 * 256 * 9x^2 = 103680x^2.
So, the first term is 1024, and the third term is 103680x^2.
To find the terms in the expansion of (2-3x)^10, we can use the binomial theorem. According to the binomial theorem, the k-th term in the expansion of (a + b)^n can be found using the formula:
Term(k) = C(n, k-1) * (a)^(n-k+1) * (b)^(k-1)
Where C(n, k-1) represents the binomial coefficient, which can be calculated as C(n, k-1) = n! / (k-1)! * (n - (k - 1))!
For the given expression (2-3x)^10, we have a = 2, b = -3x, and n = 10. We can calculate the desired terms as follows:
First term (k = 1):
Term(1) = C(10, 1-1) * (2)^(10-1+1) * (-3x)^(1-1)
= C(10, 0) * 2^10 * (-3x)^0
= 1 * 2^10 * (-3^0) [Any number raised to the power of 0 is 1]
= 1024
Third term (k = 3):
Term(3) = C(10, 3-1) * (2)^(10-3+1) * (-3x)^(3-1)
= C(10, 2) * 2^8 * (-3x)^2
= (10! / 8! * (10 - 8)!) * 2^8 * (-3x)^2
= (10 * 9 / 2 * 1) * 2^8 * (-3^2 * x^2)
= 45 * 2^8 * 9 * x^2
= 90 * 256 * 9 * x^2
= 207,360 * 9 * x^2
= 1,866,240x^2
Therefore, the first term in the expansion of (2-3x)^10 is 1024, and the third term is 1,866,240x^2.