QUESTION 3Find The 4th Term In The Expansion Of $(x + 2y)^{\text{?}}$.QUESTION 3Find The Coefficient Of $x^4$ In The Expansion Of $\left(1+\frac{1}{2} X\right)^{4}$.

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Introduction

Binomial expansion is a mathematical technique used to expand expressions of the form (a+b)n(a + b)^n, where aa and bb are constants or variables, and nn is a positive integer. This technique is essential in algebra, calculus, and other branches of mathematics. In this article, we will explore how to find the 4th term in the expansion of (x+2y)n(x + 2y)^n and the coefficient of x4x^4 in the expansion of (1+12x)4\left(1+\frac{1}{2} x\right)^{4}.

The Binomial Theorem

The binomial theorem states that for any positive integer nn, the expansion of (a+b)n(a + b)^n is given by:

(a+b)n=(n0)anb0+(n1)an−1b1+(n2)an−2b2+⋯+(nn−1)a1bn−1+(nn)a0bn(a + b)^n = \binom{n}{0} a^n b^0 + \binom{n}{1} a^{n-1} b^1 + \binom{n}{2} a^{n-2} b^2 + \cdots + \binom{n}{n-1} a^1 b^{n-1} + \binom{n}{n} a^0 b^n

where (nk)\binom{n}{k} is the binomial coefficient, defined as:

(nk)=n!k!(n−k)!\binom{n}{k} = \frac{n!}{k!(n-k)!}

Finding the 4th Term in the Expansion of (x+2y)n(x + 2y)^n

To find the 4th term in the expansion of (x+2y)n(x + 2y)^n, we need to find the term that corresponds to the 4th power of xx and 2y2y. Using the binomial theorem, we can write:

(x+2y)n=(n0)xn(2y)0+(n1)xn−1(2y)1+(n2)xn−2(2y)2+⋯+(nn−1)x1(2y)n−1+(nn)x0(2y)n(x + 2y)^n = \binom{n}{0} x^n (2y)^0 + \binom{n}{1} x^{n-1} (2y)^1 + \binom{n}{2} x^{n-2} (2y)^2 + \cdots + \binom{n}{n-1} x^1 (2y)^{n-1} + \binom{n}{n} x^0 (2y)^n

The 4th term in this expansion is:

(n3)xn−3(2y)3=(n3)xn−38y3\binom{n}{3} x^{n-3} (2y)^3 = \binom{n}{3} x^{n-3} 8y^3

To find the value of nn, we need to know the power of xx in the 4th term. Let's assume that the power of xx in the 4th term is mm. Then, we can write:

n−3=mn - 3 = m

Solving for nn, we get:

n=m+3n = m + 3

Now, we need to find the value of mm. Since the power of xx in the 4th term is mm, we can write:

m=n−3m = n - 3

Substituting this expression for mm into the equation for nn, we get:

n=(n−3)+3n = (n - 3) + 3

Simplifying this equation, we get:

n=nn = n

This equation is true for any value of nn. Therefore, we need more information to find the value of nn.

Finding the Coefficient of x4x^4 in the Expansion of (1+12x)4\left(1+\frac{1}{2} x\right)^{4}

To find the coefficient of x4x^4 in the expansion of (1+12x)4\left(1+\frac{1}{2} x\right)^{4}, we can use the binomial theorem:

(1+12x)4=(40)14(12x)0+(41)13(12x)1+(42)12(12x)2+(43)11(12x)3+(44)10(12x)4\left(1+\frac{1}{2} x\right)^{4} = \binom{4}{0} 1^4 \left(\frac{1}{2} x\right)^0 + \binom{4}{1} 1^3 \left(\frac{1}{2} x\right)^1 + \binom{4}{2} 1^2 \left(\frac{1}{2} x\right)^2 + \binom{4}{3} 1^1 \left(\frac{1}{2} x\right)^3 + \binom{4}{4} 1^0 \left(\frac{1}{2} x\right)^4

The term that corresponds to the coefficient of x4x^4 is:

(44)10(12x)4=(44)(12)4x4=1â‹…116x4=116x4\binom{4}{4} 1^0 \left(\frac{1}{2} x\right)^4 = \binom{4}{4} \left(\frac{1}{2}\right)^4 x^4 = 1 \cdot \frac{1}{16} x^4 = \frac{1}{16} x^4

Therefore, the coefficient of x4x^4 in the expansion of (1+12x)4\left(1+\frac{1}{2} x\right)^{4} is 116\frac{1}{16}.

Conclusion

In this article, we have explored how to find the 4th term in the expansion of (x+2y)n(x + 2y)^n and the coefficient of x4x^4 in the expansion of (1+12x)4\left(1+\frac{1}{2} x\right)^{4}. We have used the binomial theorem to expand these expressions and find the desired terms. The binomial theorem is a powerful tool in algebra and calculus, and it has many applications in mathematics and other fields.

References

Further Reading

Introduction

Binomial expansion is a mathematical technique used to expand expressions of the form (a+b)n(a + b)^n, where aa and bb are constants or variables, and nn is a positive integer. This technique is essential in algebra, calculus, and other branches of mathematics. In this article, we will answer some frequently asked questions about binomial expansion.

Q: What is the binomial theorem?

A: The binomial theorem is a mathematical formula that describes the expansion of expressions of the form (a+b)n(a + b)^n, where aa and bb are constants or variables, and nn is a positive integer.

Q: How do I use the binomial theorem to expand an expression?

A: To use the binomial theorem to expand an expression, you need to follow these steps:

  1. Identify the values of aa, bb, and nn in the expression.
  2. Use the binomial theorem formula to write the expansion of the expression.
  3. Simplify the expression by combining like terms.

Q: What is the binomial coefficient?

A: The binomial coefficient is a number that appears in the binomial theorem formula. It is defined as:

(nk)=n!k!(n−k)!\binom{n}{k} = \frac{n!}{k!(n-k)!}

where nn is a positive integer and kk is a non-negative integer.

Q: How do I find the value of the binomial coefficient?

A: To find the value of the binomial coefficient, you can use the formula:

(nk)=n!k!(n−k)!\binom{n}{k} = \frac{n!}{k!(n-k)!}

You can also use a calculator or a computer program to find the value of the binomial coefficient.

Q: What is the difference between the binomial theorem and the binomial coefficient?

A: The binomial theorem is a mathematical formula that describes the expansion of expressions of the form (a+b)n(a + b)^n, while the binomial coefficient is a number that appears in the binomial theorem formula.

Q: How do I use the binomial theorem to find the coefficient of a term in an expansion?

A: To use the binomial theorem to find the coefficient of a term in an expansion, you need to follow these steps:

  1. Identify the term you want to find the coefficient of.
  2. Use the binomial theorem formula to write the expansion of the expression.
  3. Identify the binomial coefficient that corresponds to the term you want to find the coefficient of.
  4. Simplify the expression by combining like terms.

Q: What is the relationship between the binomial theorem and the binomial distribution?

A: The binomial theorem is a mathematical formula that describes the expansion of expressions of the form (a+b)n(a + b)^n, while the binomial distribution is a probability distribution that describes the number of successes in a fixed number of independent trials.

Q: How do I use the binomial theorem to solve problems in algebra and calculus?

A: To use the binomial theorem to solve problems in algebra and calculus, you need to follow these steps:

  1. Identify the problem you want to solve.
  2. Use the binomial theorem formula to write the expansion of the expression.
  3. Simplify the expression by combining like terms.
  4. Use the resulting expression to solve the problem.

Conclusion

In this article, we have answered some frequently asked questions about binomial expansion. We have discussed the binomial theorem, the binomial coefficient, and how to use the binomial theorem to expand expressions and find the coefficient of a term in an expansion. We have also discussed the relationship between the binomial theorem and the binomial distribution, and how to use the binomial theorem to solve problems in algebra and calculus.

References

Further Reading