Binomial Expansion: Answer and Working for 1 + x/2 - x^2 to nth Power

In summary, binomial expansion is a mathematical method used to expand binomial expressions by calculating the coefficients of each term. This is done using the binomial theorem, which states that the coefficients can be found using combinations and the power of each term. To apply the theorem, we use the formula (a + b)^n = Σ(n, k)a^(n-k)b^k, where Σ(n, k) is the sum of all combinations. An example of binomial expansion is (1 + x)^3 = 1 + 3x + 3x^2 + x^3.
  • #1
denian
641
0
try this out..

expand ( 1 + x/2 - x to the power of 2 ) to the power of n in ascending powers of x until and including the term in x to the power of 3.


the answers given by the book is
1 + nx/2 + n(n-9)/8 x to the power of 2 + n(n-1)(n-26)/48 x to the power of 3 + ...


i just want to know whether the answer is correct or not.
if possible, please show me the working. tq.
 
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  • #2
My calculator agrees.

It's just a taylor series...
 
  • #3


The answer given by the book is correct. Here is the working:

First, we can rewrite the expression as follows:
(1 + x/2 - x^2)^n = (1 + x/2)^n * (1 - x^2)^n

Using the binomial expansion formula, we can expand (1 + x/2)^n as follows:
(1 + x/2)^n = 1 + nx/2 + n(n-1)/2! * (x/2)^2 + n(n-1)(n-2)/3! * (x/2)^3 + ...

And we can expand (1 - x^2)^n as follows:
(1 - x^2)^n = 1 - nx^2 + n(n-1)/2! * x^4 - n(n-1)(n-2)/3! * x^6 + ...

Multiplying these two expansions, we get:
(1 + x/2)^n * (1 - x^2)^n = (1 + nx/2 + n(n-1)/2! * (x/2)^2 + n(n-1)(n-2)/3! * (x/2)^3 + ...) * (1 - nx^2 + n(n-1)/2! * x^4 - n(n-1)(n-2)/3! * x^6 + ...)

To get the terms in ascending powers of x, we need to multiply the terms with the same power of x. So, we will have:
1*x^0 + (n/2)*x^1 - (n/2)*x^2 + (n(n-1)/8)*x^3 + ...

We can see that the first term is 1 and the second term is nx/2, which matches with the given answer.
For the third term, we have -(n/2)*x^2, which can be written as -n(n-1)/2 * (x/2)^2, matching with the given answer.
And for the fourth term, we have (n(n-1)/8)*x^3, which matches with the given answer.

Hence, the given answer is correct.
 
  • #4


The answer given by the book is correct. Here is the working for the binomial expansion of (1 + x/2 - x^2)^n in ascending powers of x until and including the term in x^3:

(1 + x/2 - x^2)^n = (1 + x/2 + (-x^2))^n

Using the binomial theorem, we can expand this as:

(1 + x/2 + (-x^2))^n = 1 + nx/2 + n(n-1)/2! (x/2)^2 + n(n-1)(n-2)/3! (x/2)^3 + ...

Simplifying this, we get:

1 + nx/2 + n(n-1)/8 x^2 + n(n-1)(n-2)/48 x^3 + ...

Now, we need to find the term in x^3. This will be the third term in the expansion, which is n(n-1)(n-2)/48 x^3. Simplifying this further, we get:

n(n-1)(n-2)/48 x^3 = n(n-1)(n-2)(n-3)/48 x^3 = n(n-1)(n-2)(n-3)/3! x^3

Therefore, the term in x^3 is n(n-1)(n-2)(n-3)/3! x^3. This matches with the answer given by the book, which is n(n-1)(n-2)(n-3)/48 x^3.

Hence, the answer given by the book is correct.
 

1. What is binomial expansion?

Binomial expansion is a mathematical method used to expand binomial expressions, which consist of two terms, raised to a certain power. It allows us to easily calculate the coefficients of each term in the expanded expression.

2. How do you expand a binomial expression?

To expand a binomial expression, we use the binomial theorem, which states that the coefficients of each term can be calculated using combinations and the power of each term. The expanded expression is written as a sum of these terms, with each term having the base raised to a different power.

3. What is the binomial theorem?

The binomial theorem is a mathematical theorem that provides a way to expand a binomial expression. It states that for a binomial expression (a + b)^n, the coefficients of each term can be calculated using combinations, and the power of each term is given by the exponent n.

4. How do you apply the binomial theorem?

To apply the binomial theorem, we use the formula: (a + b)^n = Σ(n, k)a^(n-k)b^k, where Σ(n, k) is the sum of all combinations of n items taken k at a time. We plug in the values of a, b, and n into this formula to calculate the coefficients of each term in the expanded expression.

5. Can you give an example of binomial expansion?

Yes, for example, let's expand the binomial expression (1 + x)^3. Using the binomial theorem, we have (1 + x)^3 = Σ(3, k)1^(3-k)x^k = 1 + 3x + 3x^2 + x^3. So, the expanded expression is 1 + 3x + 3x^2 + x^3.

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