In the Binomial expansion of (1+ax)k where a and k are non-zero constants, the coefficient of x is 8 and the coefficient of x2 is 30 a.Determine the value of a and k b.Find the coefficient of x3

a. To determine the value of a and k, we can use the binomial expansion formula. The coefficient of x in the expansion is given as 8, which means that the term involving x is 8x. Similarly, the coefficient of x^2 is given as 30, which means that the term involving x^2 is 30x^2.

Using the binomial expansion formula, we can write the expansion as:

(1+ax)^k = 1 + k(ax) + (k(k-1)/2)(ax)^2 + ...

Comparing this with the given coefficients, we have:

8x = k(ax) ----(1) 30x^2 = (k(k-1)/2)(ax)^2 ----(2)

From equation (1), we can see that k(ax) = 8x. Since a and k are non-zero constants, we can divide both sides of the equation by x to get:

k(a) = 8 ----(3)

From equation (2), we can see that (k(k-1)/2)(ax)^2 = 30x^2. Again, dividing both sides by x^2, we get:

(k(k-1)/2)(a)^2 = 30 ----(4)

Now, we have two equations (3) and (4) with two unknowns (a and k). We can solve these equations simultaneously to find the values of a and k.

From equation (3), we have k(a) = 8. Substituting the value of k from equation (3) into equation (4), we get:

(8(8-1)/2)(a)^2 = 30

Simplifying this equation, we have:

28a^2 = 30

Dividing both sides by 28, we get:

a^2 = 30/28

Taking the square root of both sides, we get:

a = ±√(30/28)

Simplifying further, we have:

a = ±√(15/14)

Therefore, the value of a can be either √(15/14) or -√(15/14).

To find the value of k, we can substitute the value of a into equation (3):

k(√(15/14)) = 8

Simplifying this equation, we have:

k = 8/√(15/14)

Simplifying further, we have:

k = 8√(14/15)

Therefore, the value of k is 8√(14/15).

b. To find the coefficient of x^3, we can use the binomial expansion formula again. The coefficient of x^3 can be found by considering the term involving x^3 in the expansion.

Using the binomial expansion formula, we have:

(1+ax)^k = 1 + k(ax) + (k(k-1)/2)(ax)^2 + (k(k-1)(k-2)/6)(ax)^3 + ...

The coefficient of x^3 is given by (k(k-1)(k-2)/6)(ax)^3.

Substituting the values of a and k that we found earlier, we have:

(k(k-1)(k-2)/6)(ax)^3 = (8√(14/15)(8√(14/15)-1)(8√(14/15)-2)/6)(ax)^3

Simplifying this expression will give us the coefficient of x^3.

a. To determine the value of a and k, we can use the binomial expansion formula. The coefficient of x in the expansion is given as 8, which means that the term involving x is 8x. Similarly, the coefficient of x^2 is given as 30, which means that the term involving x^2 is 30x^2.

Using the binomial expansion formula, we can write the expansion as:

(1+ax)^k = 1 + k(ax) + (k(k-1)/2)(ax)^2 + ...

Comparing this with the given coefficients, we have:

8x = k(ax) ----(1) 30x^2 = (k(k-1)/2)(ax)^2 ----(2)

From equation (1), we can see that k(ax) = 8x. Since a and k are non-zero constants, we can divide both sides of the equation by x to get:

k(a) = 8 ----(3)

From equation (2), we can see that (k(k-1)/2)(ax)^2 = 30x^2. Again, dividing both sides by x^2, we get:

(k(k-1)/2)(a)^2 = 30 ----(4)

Now, we have two equations (3) and (4) with two unknowns (a and k). We can solve these equations simultaneously to find the values of a and k.

From equation (3), we have k(a) = 8. Substituting the value of k from equation (3) into equation (4), we get:

(8(8-1)/2)(a)^2 = 30

Simplifying this equation, we have:

28a^2 = 30

Dividing both sides by 28, we get:

a^2 = 30/28

Taking the square root of both sides, we get:

a = ±√(30/28)

Simplifying further, we have:

a = ±√(15/14)

Therefore, the value of a can be either √(15/14) or -√(15/14).

Substituting the value of a into equation (3), we can find the value of k:

k(√(15/14)) = 8 or k(-√(15/14)) = 8

Solving these equations, we find:

k = 8/√(15/14) or k = -8/√(15/14)

Simplifying further, we have:

k = 8√(14/15) or k = -8√(14/15)

Therefore, the values of a and k are:

a = ±√(15/14) k = 8√(14/15) or k = -8√(14/15)

b. To find the coefficient of x^3, we can use the binomial expansion formula. The coefficient of x^3 is given by the term involving (ax)^3, which can be written as (k(k-1)(k-2)/6)(ax)^3.

Substituting the values of a and k that we found in part a, we can calculate the coefficient of x^3.