Approximating 1.01^20 Using the Binomial Theorem to Four Decimal Places
Introduction to the Binomial Theorem
The binomial theorem is a powerful mathematical tool used to expand powers of a binomial expression. A binomial is any expression with two terms, such as ( (a b) ). One of its most useful applications is in approximating powers of expressions that are close to 1. In this article, we will explore how to use the binomial theorem to approximate ( 1.01^{20} ) to four decimal places. This is a common type of problem where the binomial theorem can be especially handy.
Step-by-Step Calculation Using the Binomial Theorem
To approximate ( 1.01^{20} ), we can apply the binomial theorem directly. Recall the formula for the binomial expansion of ( (1 x)^n ) is:
( (1 x)^n 1 nx frac{n(n-1)}{2}x^2 frac{n(n-1)(n-2)}{6}x^3 cdots )
Where ( n ) is a real number (in this case, 20) and ( x ) is a small real number (in this case, 0.01).
Let's apply this to our problem:
( (1.01)^{20} (1 0.01)^{20} )
Using the binomial theorem:
( (1 0.01)^{20} 1 20(0.01) frac{20 cdot 19}{2}(0.01)^2 frac{20 cdot 19 cdot 18}{6}(0.01)^3 cdots )
Let's calculate the first few terms of this series:
( 1 ) ( 20(0.01) 0.2 ) ( frac{20 cdot 19}{2}(0.01)^2 190 cdot 0.0001 0.019 ) ( frac{20 cdot 19 cdot 18}{6}(0.01)^3 1140 cdot 0.000001 0.00114 ) ( cdots ) Continuing the expansion, subsequent terms will be much smaller and can be disregarded for our approximation.Now, let's sum the important terms:
( 1 0.2 0.019 0.00114 cdots approx 1 0.2 0.019 0.00114 1.22014 )
Therefore, ( 1.01^{20} ) to four decimal places is approximately ( 1.2202 ).
Why Use the Binomial Theorem?
The binomial theorem is useful in many practical scenarios, especially when dealing with small differences or small changes. For example, it can be used in finance to approximate compound interest, in physics to understand small perturbations, and in all sorts of engineering and scientific calculations where precise approximations are needed but full calculations might be too complex.
Conclusion
The binomial theorem is a crucial tool in mathematics, providing powerful methods to approximate various expressions. By understanding and applying it correctly, one can easily calculate complex expressions like ( 1.01^{20} ) with relative ease and precision. Whether you are a student, a researcher, or a professional dealing with quantitative data, mastering the binomial theorem can enhance your problem-solving skills.