Question

1. evaluate: s4 = ∑_{k = 1}^{4}2(3^{n - 1})
2. determine the length of x in the triangle. give your answer to two decimal places. you may use the desmos scientific calculator: https://www.desmos.com/scientific.

Explanation:

Step1: Expand the sum for $n = 1,2,3,4$

When $n = 1$, $2(3^{1 - 1})=2(3^{0})=2\times1 = 2$.
When $n = 2$, $2(3^{2 - 1})=2\times3^{1}=2\times3 = 6$.
When $n = 3$, $2(3^{3 - 1})=2\times3^{2}=2\times9 = 18$.
When $n = 4$, $2(3^{4 - 1})=2\times3^{3}=2\times27 = 54$.

Step2: Calculate the sum

$s_{4}=2 + 6+18 + 54=80$.

for question 17:

Step1: Use the sine - function in a right - triangle

In a right - triangle, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. Here, $\theta = 20^{\circ}$ and the opposite side to the angle $\theta$ is $12$ and the hypotenuse is $x$. So, $\sin(20^{\circ})=\frac{12}{x}$.

Step2: Solve for $x$

We can rewrite the equation as $x=\frac{12}{\sin(20^{\circ})}$. Since $\sin(20^{\circ})\approx0.3420$, then $x=\frac{12}{0.3420}\approx35.09$.

Answer:

$80$