Answer:
2065.5
Step-by-step explanation:
You have to approximate the following calculation 3.5^6
The binomial theorem is given by:
[tex](a+b)^n=\Sigma_{k=0}^{k=n}\left[\begin{array}{c}n&k\end{array}\right] a^{n-k}b^k[/tex] (1)
You can express the number 3as follow:
[tex]3.5^6=(3+0.5)^6=(3+\frac{1}{2})^6[/tex] (2)
by comparing the equation (1) with the equation (2) you have
a = 3
b = 1/2
To calculate the first three terms of the binomial theorem you use k=3. You replace the values of a, b, n and k in the equation (1):
[tex](3+\frac{1}{2})^6=\Sigma_{k=0}^{k=6}\left[\begin{array}{c}6&k\end{array}\right] (3)^{6-k}b^k[/tex]
You only take the first three terms:
[tex](3+\frac{1}{2}^6)\approx(1)(3)^6(\frac{1}{2})^0+(6)(3)^5(\frac{1}{2})^1+\frac{6\cdot 5}{1\cdot 2}(3)^4(\frac{1}{2})^2\\\\(3+\frac{1}{2}^6)\approx729+729+\frac{1215}{2}=2065.5[/tex]
By using the first three terms of bynomial theorem you obtain 2065.5
