Binomial Expansion: Finding the 4th Term
The binomial theorem provides a formula for expanding expressions of the form $(x+y)^n$, where $n$ is a non-negative integer. The expansion results in a sum of terms, each with a specific coefficient and powers of $x$ and $y$.
Understanding the Pattern
To find the 4th term of a binomial expansion, we can utilize the pattern of the binomial coefficients. Each term in the expansion follows the following format:
(1) The coefficient is determined by the binomial coefficient, calculated using the formula:
nCr = n! / (r! * (n-r)!), where 'n' is the power of the binomial and 'r' represents the term number (starting from 0).
(2) The power of 'x' decreases from 'n' to 0, while the power of 'y' increases from 0 to 'n'.
Finding the 4th Term
To find the 4th term, we consider the following:
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Term Number: Since the terms start from 0, the 4th term corresponds to r = 3.
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Binomial Coefficient: We calculate the coefficient using the formula above:
nCr = n! / (r! * (n-r)!)
nC3 = n! / (3! * (n-3)!)
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Powers of 'x' and 'y': The power of 'x' is (n - r) and the power of 'y' is r. For the 4th term:
- Power of 'x': n - 3
- Power of 'y': 3
Putting it Together
Therefore, the 4th term of the binomial expansion of $(x+y)^n$ is:
(n! / (3! * (n-3)!)) * x^(n-3) * y^3
Example
Let's find the 4th term of the expansion of (x + 2y)^5.
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n = 5, r = 3
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Binomial Coefficient:
5C3 = 5! / (3! * 2!) = 10
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Powers:
- x^(5-3) = x^2
- (2y)^3 = 8y^3
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Putting it Together:
The 4th term is 10 * x^2 * 8y^3 = 80x^2y^3
Conclusion
Using the binomial theorem and understanding the pattern of coefficients and powers, we can easily find any specific term in the expansion of a binomial expression. The above steps provide a systematic approach to calculate the 4th term of the binomial expansion.
