Determine the coefficient of x^15 in the expansion (2x-3x^2)^10
ii)Find the coefficient of x^-8 in the expansion (x^2 +1/x^3)^16
General term:
term(n+1) = C(10,n) (2x)^n (-3x^2)^(10-n)
= C(10,n) (2^n) x^n (-3)^(10-n) (x^2)^(10-n)
= C(10,n) 2^n (-3)^(10-n) x^(20-n)
we want x^(20-n) = x^15
20-n = 15
n = 5
So the coefficient is C(10,5)(2^5)(-3)^5
= 252(32)(-243) = 1,959,552
check my arithmetic
do the 2nd part the same way.
Sorry but the part where u said (x^2)^{10-n)=x^20 -n
You should have been 100% sure that it was -1959552 looking at
my 252(32)(-243)
regarding you second concern:
notice that there were 2 factors containing the base x
(x^n)(x^2)^(10-n)
= (x^n)(x^(20 - 2n) )
= x^(n + 20 - 2n)
= x^(20-n)
To find the coefficient of a particular term in a binomial expansion, we can use the Binomial Theorem. The Binomial Theorem states that for a binomial expression (a + b)^n, the coefficient of the term containing a^m * b^k can be found using the formula:
Coefficient = nCm * a^m * b^k
Where nCm (also known as "n choose m") represents the number of ways we can choose m items from a set of n items, and is given by the formula:
nCm = n! / (m! * (n-m)!)
Now let's solve the problem using the above formula:
i) To find the coefficient of x^15 in the expansion (2x - 3x^2)^10, we need to determine the values of m and k that satisfy the equation m + 2k = 15. Since the highest power of x in the binomial expression (2x - 3x^2) is x^2, we have m ≤ 2 and k = (15 - m)/2.
Substituting the values of m and k into the formula, we have:
Coefficient = 10Cm * (2x)^m * (-3x^2)^k = 10Cm * 2^m * (-3)^k * x^(2k + m)
To find the coefficient of x^15, we must have 2k + m = 15:
2k + m = 15
2(15 - m)/2 + m = 15
15 - m + m = 15
15 = 15
This means that we have a solution for any value of m from 0 to 2. Thus, the coefficient of x^15 in the expansion (2x - 3x^2)^10 is the sum of the coefficients of the terms with m = 0, 1, and 2.
Coefficient(x^15) = 10C0 * 2^0 * (-3)^5 * x^(2*5 + 0)
+ 10C1 * 2^1 * (-3)^4 * x^(2*4 + 1)
+ 10C2 * 2^2 * (-3)^3 * x^(2*3 + 2)
Using the formula for nCm and simplifying the expression, we can calculate the individual terms and find the coefficient of x^15.
ii) To find the coefficient of x^-8 in the expansion (x^2 + 1/x^3)^16, we need to determine the values of m and k that satisfy the equation 2m - 3k = -8. Since the highest power of x in the binomial expression (x^2 + 1/x^3) is x^2, we have m ≤ 2 and k = (-8 - 2m)/3.
Substituting the values of m and k into the formula, we have:
Coefficient = 16Cm * (x^2)^m * (1/x^3)^k = 16Cm * x^(2m - 3k) * x^(-3k)
To find the coefficient of x^-8, we must have 2m - 3k = -8:
2m - 3k = -8
2m - 3((-8 - 2m)/3) = -8
2m + 8 + 6m = -8
8m + 8 = -8
8m = -16
m = -2
This means that we need to find the coefficient of the x^(-3k) term in the expansion. Substituting m = -2 into the formula, we have:
Coefficient(x^-8) = 16C(-2) * x^(2*(-2) - 3k)
Using the formula for nCm, the coefficient becomes:
Coefficient(x^-8) = 16C(-2) * x^-4 * x^(-3k)
Simplifying the expression, we can calculate the individual terms and find the coefficient of x^-8.