Question

Use the first three terms of the binomial theorem to approximate 3.5^6.

Answer 1

Answer:

2065.5

Step-by-step explanation:

You have to approximate the following calculation 3.5^6

The binomial theorem is given by:

$(a+b)^n=\Sigma_{k=0}^{k=n}\left[\begin{array}{c}n&k\end{array}\right] a^{n-k}b^k$      (1)

You can express the number 3as follow:

$3.5^6=(3+0.5)^6=(3+\frac{1}{2})^6$          (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):

$(3+\frac{1}{2})^6=\Sigma_{k=0}^{k=6}\left[\begin{array}{c}6&k\end{array}\right] (3)^{6-k}b^k$

You only take the first three terms:

$(3+\frac{1}{2}^6)\approx(1)(3)^6(\frac{1}{2})^0+(6)(3)^5(\frac{1}{2})^1+\frac{6\cdot 5}{1\cdot 2}(3)^4(\frac{1}{2})^2\\\\(3+\frac{1}{2}^6)\approx729+729+\frac{1215}{2}=2065.5$

By using the first three terms of bynomial theorem you obtain 2065.5