
If the binomial expansion of is given as , then find the value of a and n?
(a) 2, 4
(b) 2, 3
(c) 3, 6
(d) 1, 2.
Answer
509.7k+ views
Hint: We can see that the binomial expansion is similar to . So, we expand and compare the coefficients of x and on both sides to get the equations. We solve the equation to get the value of one of the variables. Using the obtained value of the variable, we find the value of the other variable by using one of the equations.
Complete step by step answer:
Given that we have a binomial expansion , given as . We need to find the values of ‘a’ and ‘n’.
The binomial expansion is similar to the expansion .
We know that the binomial expansion of is given as:
---(1).
We use equation (1), to expand the binomial expansion .
So, the binomial expansion of is:
---(2).
According to the problem, the binomial expansion of is given as ---(3).
Using equations (2) and (3), we get
= .
= ---(4).
Comparing coefficients of ‘x’ terms on both sides, we get .
We know that the value of is given as .
So, we have got .
We have got ---(5).
Equating coefficients of on both sides from equation (4), we get .
We have got .
We have got .
We have got .
We have got ---(6).
We substitute the value obtained from equation (5) in equation (6).
We have got .
We have got .
We have got .
We have got .
We have got a = 2.
Now we substitute the obtained value of ‘a’ in the equation (5) to find the value of ‘n’.
We have got .
We have got .
We have got n = 4.
∴ The values of ‘a’ and ‘n’ is ‘2’ and ‘4’.
So, the correct answer is “Option A”.
Note: We need to make sure that the order of the answers given in options are for ‘a’ and ‘n’. Whenever we get the problems involving binomial expansion, we expand it first to solve. Similarly, we can expect to solve the problems that contain more than two variables and fractional powers.
Complete step by step answer:
Given that we have a binomial expansion
The binomial expansion
We know that the binomial expansion of
We use equation (1), to expand the binomial expansion
So, the binomial expansion of
According to the problem, the binomial expansion of
Using equations (2) and (3), we get
Comparing coefficients of ‘x’ terms on both sides, we get
We know that the value of
So, we have got
We have got
Equating coefficients of
We have got
We have got
We have got
We have got
We substitute the value
We have got
We have got
We have got
We have got
We have got a = 2.
Now we substitute the obtained value of ‘a’ in the equation (5) to find the value of ‘n’.
We have got
We have got
We have got n = 4.
∴ The values of ‘a’ and ‘n’ is ‘2’ and ‘4’.
So, the correct answer is “Option A”.
Note: We need to make sure that the order of the answers given in options are for ‘a’ and ‘n’. Whenever we get the problems involving binomial expansion, we expand it first to solve. Similarly, we can expect to solve the problems that contain more than two variables and fractional powers.
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