Binomial Expansion Of (1 + (1/4)x)^10 And Approximation Of (41/40)^10

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Introduction

In the realm of mathematics, the binomial theorem stands as a cornerstone for expanding expressions of the form (a + b)^n, where n is a non-negative integer. This theorem provides a systematic method for determining the coefficients and terms that arise when such expressions are raised to a power. In this article, we delve into the intricacies of binomial expansion, focusing on the specific case of (1 + (1/4)x)^10. Our primary objective is to determine the binomial expansion of this expression in ascending powers of x, up to and including the term x^3. Furthermore, we aim to leverage this expansion to approximate the value of (41/40)^10, showcasing the practical applications of the binomial theorem.

Binomial Theorem: A Foundation for Expansion

The binomial theorem provides a powerful formula for expanding expressions of the form (a + b)^n, where n is a non-negative integer. The theorem states that:

(a + b)^n = Σ (n choose k) * a^(n-k) * b^k

where the summation (Σ) is taken from k = 0 to k = n, and (n choose k) represents the binomial coefficient, which is defined as:

(n choose k) = n! / (k! * (n-k)!)

Here, n! denotes the factorial of n, which is the product of all positive integers up to n. The binomial coefficient (n choose k) represents the number of ways to choose k objects from a set of n objects, without regard to order. This concept is fundamental in combinatorics and probability theory.

In the context of our problem, we have a = 1, b = (1/4)x, and n = 10. Applying the binomial theorem, we can expand (1 + (1/4)x)^10 as follows:

(1 + (1/4)x)^10 = (10 choose 0) * 1^10 * ((1/4)x)^0 + (10 choose 1) * 1^9 * ((1/4)x)^1 + (10 choose 2) * 1^8 * ((1/4)x)^2 + (10 choose 3) * 1^7 * ((1/4)x)^3 + ...

To determine the expansion up to the term x^3, we need to calculate the binomial coefficients and simplify the terms corresponding to k = 0, 1, 2, and 3.

Determining the Binomial Expansion up to x^3

Let's calculate the binomial coefficients and simplify the terms in the expansion:

  • For k = 0: (10 choose 0) = 10! / (0! * 10!) = 1. The term is 1 * 1^10 * ((1/4)x)^0 = 1.
  • For k = 1: (10 choose 1) = 10! / (1! * 9!) = 10. The term is 10 * 1^9 * ((1/4)x)^1 = (10/4)x = (5/2)x.
  • For k = 2: (10 choose 2) = 10! / (2! * 8!) = (10 * 9) / (2 * 1) = 45. The term is 45 * 1^8 * ((1/4)x)^2 = 45 * (1/16)x^2 = (45/16)x^2.
  • For k = 3: (10 choose 3) = 10! / (3! * 7!) = (10 * 9 * 8) / (3 * 2 * 1) = 120. The term is 120 * 1^7 * ((1/4)x)^3 = 120 * (1/64)x^3 = (15/8)x^3.

Therefore, the binomial expansion of (1 + (1/4)x)^10 up to the term x^3 is:

(1 + (1/4)x)^10 ≈ 1 + (5/2)x + (45/16)x^2 + (15/8)x^3

This expansion provides an approximation of the original expression for values of x close to zero. The accuracy of the approximation decreases as the value of x increases.

Leveraging the Expansion to Approximate (41/40)^10

Now, let's utilize the obtained expansion to approximate the value of (41/40)^10. We can rewrite (41/40)^10 as (1 + 1/40)^10. By comparing this expression with (1 + (1/4)x)^10, we can observe that x = 1/10. Substituting x = 1/10 into our binomial expansion, we get:

(1 + (1/4)(1/10))^10 ≈ 1 + (5/2)(1/10) + (45/16)(1/10)^2 + (15/8)(1/10)^3

Simplifying the expression:

(1 + 1/40)^10 ≈ 1 + 1/4 + 45/1600 + 15/8000

To obtain a common denominator, we can rewrite the terms as:

(1 + 1/40)^10 ≈ 1 + 1000/4000 + 112.5/4000 + 7.5/4000

Combining the terms:

(1 + 1/40)^10 ≈ (4000 + 1000 + 112.5 + 7.5) / 4000

(1 + 1/40)^10 ≈ 5120 / 4000

(1 + 1/40)^10 ≈ 1.28

Therefore, using the binomial expansion, we approximate the value of (41/40)^10 to be approximately 1.28.

Accuracy and Limitations of the Approximation

It's crucial to acknowledge that the binomial expansion provides an approximation, and the accuracy of this approximation depends on the value of x and the number of terms included in the expansion. In our case, we only considered terms up to x^3. Including more terms would generally improve the accuracy of the approximation, especially for larger values of x. However, for values of x significantly greater than zero, the approximation may become less reliable.

In this specific example, x = 1/10, which is relatively small. Therefore, the approximation is reasonably accurate. However, if we were to approximate (1 + (1/4)x)^10 for a larger value of x, such as x = 1, the approximation would be less accurate. This is because the higher-order terms in the binomial expansion become more significant as x increases.

To obtain a more accurate result, one could include more terms in the binomial expansion or utilize alternative methods, such as direct calculation using a calculator or computer software. However, the binomial expansion provides a valuable tool for approximating expressions, particularly when dealing with fractional or complex powers.

Conclusion

In this article, we successfully determined the binomial expansion of (1 + (1/4)x)^10 up to the term x^3. This expansion provides a polynomial approximation of the original expression, which can be used for various applications, such as approximating values for specific values of x. We then demonstrated the practical application of this expansion by approximating the value of (41/40)^10. By substituting x = 1/10 into the expansion, we obtained an approximate value of 1.28.

While the binomial expansion offers a powerful approximation technique, it's essential to understand its limitations. The accuracy of the approximation depends on the value of x and the number of terms included in the expansion. For larger values of x, including more terms or employing alternative methods may be necessary to achieve a more accurate result.

The binomial theorem and its applications, such as binomial expansion, are fundamental concepts in mathematics with wide-ranging applications in various fields, including statistics, probability, and physics. Understanding these concepts provides a valuable toolkit for problem-solving and approximation in diverse contexts.

Further Exploration

To further explore the topic of binomial expansion and its applications, consider the following avenues:

  • Investigate the binomial theorem for non-integer exponents.
  • Explore the connection between binomial expansion and combinatorics.
  • Apply binomial expansion to solve problems in probability and statistics.
  • Utilize binomial expansion to approximate functions and solve differential equations.

By delving deeper into these areas, you can gain a more comprehensive understanding of the power and versatility of the binomial theorem and its applications.