Problem 57 Approximate \((1.02)^{8}\) by ev... [FREE SOLUTION]

Chapter 6: Problem 57

Approximate \((1.02)^{8}\) by evaluating the first three terms of \((1+0.02)^{s}\)

Short Answer

Expert verified

The approximation of \((1.02)^{8}\) by evaluating the first three terms of \((1+0.02)^{8}\) using the Binomial Theorem is 1.1712.

Step by step solution

Understand the Binomial Theorem

The Binomial Theorem states that \((a+b)^{n} = a^n + n*a^{n-1}*b^1 + n(n-1)/2*a^{n-2}*b^2 + ... up to b^n\). In your case, \(a = 1\), \(b = 0.02\) and \(n = 8\).

Apply the Binomial Theorem and Calculate First Three Terms

According to the Binomial Theorem, the first term is \(a^n\), which is \(1^8 = 1\). The second term is \(n*a^{n-1}*b\), which is \(8*1^{7}*0.02 = 0.16\). The third term is \(n(n-1)/2*a^{n-2}*b^2\), which is \(8*7/2*1^{6}*0.02^2 = 0.0112\).

Sum up the calculated terms

The sum of the first three terms that you've calculated is \(1 + 0.16 + 0.0112 = 1.1712\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Binomial Expansion

The binomial expansion helps us express expressions like \((a+b)^n\) in a way that鈥檚 simpler to work with. It's a powerful tool when we want to raise a binomial expression to a power. Imagine you have \((1+0.02)^8\). By using the binomial expansion, we can break it down into specific terms to make calculations easier.
Here鈥檚 how it generally works:

The first term is always \(a^n\).
The second term takes into account one \(b\) and the rest \(a\)'s.
Following terms involve more \(b\) factors and fewer \(a\)'s.

This is particularly useful for quickly approximating results without a calculator or for large powers. You only calculate as many terms as needed.

Using Approximation Techniques

Approximation techniques simplify complex calculations. In the case of binomial expressions, they help in calculating only a few terms rather than the whole expansion.
Consider approximate calculations of \((1+0.02)^8\) using the first three terms.

Why approximate?

Faster and easier calculations.
Good enough for practical purposes.
Reduces computational errors in manual calculations.

These techniques are particularly useful in real-world scenarios where an exact value isn't necessary, or when the calculation time needs to be reduced.

Exploring Mathematical Series

Mathematical series involve the sum of terms that follow a particular pattern. The binomial expansion itself is a series, where each term is derived systematically based on the theorem.
When dealing with binomial expansion, understanding it as a series helps in many ways:

You can identify patterns in the terms.
It's easier to see how terms decrease in magnitude.
Helps in visualizing and simplifying expressions.

By approximating with a series, like in our exercise, you sum only the first few terms for an easy-to-calculate answer. This notion of a series makes many complex mathematical processes more approachable and efficient.

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91影视

Approximate \((1.02)^{8}\) by evaluating the first three terms of \((1+0.02)^{s}\)

Short Answer

Step by step solution

Understand the Binomial Theorem

Apply the Binomial Theorem and Calculate First Three Terms

Sum up the calculated terms

Key Concepts

Understanding Binomial Expansion

Using Approximation Techniques

Exploring Mathematical Series

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