To solve the problem, we need to find the first four terms in the expansion of [tex]\(\left(1 - \frac{x}{10}\right)^6\)[/tex]. Then, we'll substitute [tex]\(x = 0.1\)[/tex] to approximate [tex]\((0.99)^6\)[/tex] and determine the degree of accuracy of our approximation.
Step 1: Binomial expansion
We can use the binomial theorem to expand [tex]\(\left(1 - \frac{x}{10}\right)^6\)[/tex]. According to the binomial theorem, the expansion is:
[tex]\[
\left(1 - \frac{x}{10}\right)^6 = \sum_{n=0}^{\infty} \binom{6}{n} \left(-\frac{x}{10}\right)^n
\][/tex]
The first four terms (for [tex]\(n = 0, 1, 2, 3\)[/tex]) of this expansion are:
1. For [tex]\(n = 0\)[/tex]:
[tex]\[
\binom{6}{0} \left(-\frac{x}{10}\right)^0 = 1
\][/tex]
2. For [tex]\(n = 1\)[/tex]:
[tex]\[
\binom{6}{1} \left(-\frac{x}{10}\right)^1 = 6 \left(-\frac{x}{10}\right) = -\frac{6x}{10} = -0.6x
\][/tex]
3. For [tex]\(n = 2\)[/tex]:
[tex]\[
\binom{6}{2} \left(-\frac{x}{10}\right)^2 = 15 \left(\frac{x}{10}\right)^2 = 15 \left(\frac{x^2}{100}\right) = 0.15x^2
\][/tex]
4. For [tex]\(n = 3\)[/tex]:
[tex]\[
\binom{6}{3} \left(-\frac{x}{10}\right)^3 = 20 \left(-\frac{x}{10}\right)^3 = -20 \left(\frac{x^3}{1000}\right) = -0.02x^3
\][/tex]
Therefore, the first four terms in the expansion of [tex]\(\left(1 - \frac{x}{10}\right)^6\)[/tex] are:
[tex]\[
1 - 0.6x + 0.15x^2 - 0.02x^3
\][/tex]
Step 2: Approximate [tex]\((0.99)^6\)[/tex]
To approximate [tex]\((0.99)^6\)[/tex], we substitute [tex]\( x = 0.1 \)[/tex] (since [tex]\(0.99 = 1 - 0.01 = 1 - \frac{0.1}{10}\)[/tex]) into the expansion:
[tex]\[
(0.99)^6 \approx 1 - 0.6(0.1) + 0.15(0.1^2) - 0.02(0.1^3)
\][/tex]
Calculate each term:
[tex]\[
1 - 0.6 \cdot 0.1 = 1 - 0.06 = 0.94
\][/tex]
[tex]\[
0.15 \cdot (0.1^2) = 0.15 \cdot 0.01 = 0.0015
\][/tex]
[tex]\[
-0.02 \cdot (0.1^3) = -0.02 \cdot 0.001 = -0.00002
\][/tex]
Sum these terms to get the approximate value:
[tex]\[
(0.99)^6 \approx 0.94 + 0.0015 - 0.00002 = 0.94148
\][/tex]
Step 3: Calculate the degree of accuracy
Next, we find the actual value of [tex]\((0.99)^6\)[/tex] using a calculator:
[tex]\[
(0.99)^6 \approx 0.941480149401
\][/tex]
Finally, compute the degree of accuracy of our approximation by finding the absolute difference between the actual value and the approximate value:
[tex]\[
\text{Degree of accuracy} = \left| 0.941480149401 - 0.94148 \right| = 0.000000149401 = 1.4940100001581413 \times 10^{-7}
\][/tex]
Thus, the degree of accuracy of our approximation is [tex]\(1.4940100001581413 \times 10^{-7}\)[/tex].
