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if ar is the coefficient of x^r in the expansion of (1 x x^2)^n,What is the Coefficient of x^r?

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if ar is the coefficient of x^r in the expansion of (1 x x^2)^n,What is the Coefficient of x^r?

Understanding the Coefficient of x^r in the Expansion of (1 + x + x^2)^n

When exploring the expansion of the binomial series (1 + x + x^2)^n, you might come across the term “ar,” which represents the coefficient of x^r in the series. This coefficient plays a crucial role in various mathematical applications and is a fascinating topic to delve into. Let’s explore the intricacies of this coefficient from multiple dimensions.

What is the Coefficient of x^r?

if ar is the coefficient of x^r in the expansion of (1 x x^2)^n,What is the Coefficient of x^r?

The coefficient of x^r in the expansion of (1 + x + x^2)^n is a numerical value that multiplies the term x^r in the series. It is derived from the binomial theorem, which states that the expansion of (a + b)^n can be expressed as the sum of the terms obtained by multiplying the binomial coefficients with the appropriate powers of a and b.

Binomial Theorem and Coefficients

The binomial theorem is a fundamental concept in algebra that provides a way to expand binomial expressions. It is expressed as:

(a + b)^n = 危 (nCr a^(n-r) b^r), where r ranges from 0 to n.

In our case, a = 1, b = x, and n = r. Therefore, the coefficient of x^r in the expansion of (1 + x + x^2)^n can be calculated using the binomial coefficient (nCr), which is given by:

(nCr) = n! / (r! (n-r)!) = (n (n-1) … (n-r+1)) / (r! (n-r)!) = (n (n-1) … (n-r+1)) / (r (r-1) … 1)

Calculating the Coefficient of x^r

Now that we understand the concept of the binomial coefficient, let’s calculate the coefficient of x^r in the expansion of (1 + x + x^2)^n. To do this, we need to determine the value of n and r. Once we have these values, we can use the formula mentioned above to calculate the coefficient.

For example, let’s consider the expansion of (1 + x + x^2)^4. In this case, n = 4. To find the coefficient of x^2, we need to set r = 2. Plugging these values into the formula, we get:

(4C2) = 4! / (2! (4-2)!) = (4 3) / (2 1) = 6

Therefore, the coefficient of x^2 in the expansion of (1 + x + x^2)^4 is 6.

Applications of the Coefficient of x^r

The coefficient of x^r in the expansion of (1 + x + x^2)^n has various applications in mathematics and other fields. Here are a few examples:

  • In combinatorics, the coefficient of x^r can be used to determine the number of ways to choose r items from a set of n items.

  • In probability, the coefficient of x^r can be used to calculate the probability of obtaining a specific outcome in a sequence of independent events.

  • In physics, the coefficient of x^r can be used to describe the behavior of certain physical systems, such as the motion of a particle or the propagation of waves.

Conclusion

The coefficient of x^r in the expansion of (1 + x + x^2)^n is a fascinating mathematical concept with various applications. By understanding the binomial theorem and the formula for calculating the coefficient, you can explore the intricacies of this topic and apply it to different fields. Keep exploring and expanding your knowledge in mathematics!

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