Answer:
To find the values of k and n, we can use the Binomial Theorem to expand (1 + kx)^n. The Binomial Theorem states that the kth term in the expansion of (a + b)^n is given by C(n, k) * a^(n-k) * b^k, where C(n, k) represents the binomial coefficient.
In this case, the first term in the expansion is 1, the second term is 15x, and the third term is 90x². Using the formula for the kth term, we can set up the following equations based on the given information:
First Term: C(n, 0) * (1)^(n-0) * (kx)^0 = 1
This simplifies to 1 = 1.
Second Term: C(n, 1) * (1)^(n-1) * (kx)^1 = 15x
This simplifies to C(n, 1) * kx = 15x.
Third Term: C(n, 2) * (1)^(n-2) * (kx)^2 = 90x²
This simplifies to C(n, 2) * (kx)^2 = 90x².
Let's solve these equations to find the values of k and n:
Equation 1: 1 = 1
This equation is already satisfied.
Equation 2: C(n, 1) * kx = 15x
Since kx cannot be zero (otherwise all terms will be zero), we can divide both sides by x:
C(n, 1) * k = 15
Equation 3: C(n, 2) * (kx)^2 = 90x²
We can divide both sides by x²:
C(n, 2) * k² = 90
From equation 2, we have C(n, 1) * k = 15.
Rearranging, we get C(n, 1) = 15/k.
Substituting this value into equation 3, we have (15/k) * k² = 90.
This simplifies to k² = 6.
Taking the square root of both sides, we find k = ±√6.
To find n, we can substitute the value of k into equation 2: C(n, 1) * k = 15.
For k = √6:
C(n, 1) * √6 = 15.
For k = -√6:
C(n, 1) * (-√6) = 15.
Simplifying these equations, we find two possibilities:
For k = √6:
√6 * C(n, 1) = 15.
For k = -√6:
-√6 * C(n, 1) = 15.
To solve for n, we need the value of C(n, 1), which represents the binomial coefficient. In this case, the binomial coefficient C(n, 1) corresponds to the number of ways we can choose 1 term out of n terms.
Since we have been given the first three terms in the expansion, we know that n must be at least 3. However, without further information or equations, we cannot determine the specific value of n or the value of k (whether it is √6 or -√6).
Hence, we cannot find the exact value of k and n based solely on the given information.