# Equations of conic sections, Systems of Non-linear Equations illustrate a series, differentiate a series from a sequence Determine the first five terms of each defined sequence and give it's associated series 3^{n + 1} We have to find the first five terms of the given sequence and its associated series

Question
Conic sections
Equations of conic sections, Systems of Non-linear Equations illustrate a series, differentiate a series from a sequence Determine the first five terms of each defined sequence and give it's associated series $$3^{n\ +\ 1}$$ We have to find the first five terms of the given sequence and its associated series

2021-01-20
Given that, $$n^{th}\ \text{term of the sequence is}\ 3^{n\ +\ 1}.$$ Take $$n = 0$$ $$a_{n} = 3^{n\ +\ 1}$$
$$a_{0} = 3^{0\ +\ 1} = 3$$ Therefore, the first term is 3. Take $$n = 1,$$
$$a_{1} = 3^{1\ +\ 1}$$
$$a_{1} = 3^{1\ +\ 1} = 3^{2} = 9$$ The second term is 9 Take $$n = 2,$$
$$a2 = 3^{2\ +\ 1}$$
$$= 3^{3}$$
$$a_{2} = 27$$ The third term is 27 For, $$n = 3,$$
$$a3 = 3^{3\ +\ 1}$$
$$= 3^{4}$$
$$a^{3} = 81$$ Fourth term is 81 For, $$n = 4,$$
$$a4 = 3^{4\ +\ 1}$$
$$= 3^{5}$$
$$a^{4} = 243$$ Fifth term is 243. Therefore, the first 5 terms of the sequence are $$3,\ 9,\ 27,\ 81,\ 243.$$ Step 3 The given sequence is $$3,\ 9,\ 27,\ 81,\ 243.$$ The series associated with the given sequence is $$\sum_{n=0}^{5}\ 3^{n\ +\ 1}=3\ +\ 9\ +\ 27\ +\ 81\ +\ 23$$
$$\sum_{n=0}^{5}\ 3^{n\ +\ 1}=363$$

### Relevant Questions

Equations of Conic Sections Systems of Non-linear Equations Solve eeach problem systematically 1. Find all values of m so that the graph $$2mx^{2}\ -\ 16mx\ +\ my^{2}\ +\ 7y^{2} = 2m^{2}\ -\ 18m$$ is a circle.
Recognize the equation and important characteristics of the different types of conic sections, illustrate systems of nonlinear equations, determine the solutions of system of equations (one linear and one second degree) in two variables using substitution, elimination, and graphing (in standard form), solve situational problems involving systems of non-linear equation
Write the following equation in standard form and sketch it's graph
1.$$9x^2+72x-64y^2+128y+80=0$$
2.$$y^2+56x-18y+417=0$$
3.$$x^2-10x-48y+265=0$$
4.$$x^2+4x+16y^2-128y+292=0$$
Use your some other reference source to find real-life applications of (a) linear differential equations and (b) rotation of conic sections that are different than those discussed in this section.
Give the eccentricities of conic sections with one focus at the origin along with the directrix corresponding to that focus. Find a polar equation for each conic section. $$e = 2,\ x = 4$$
The circle, ellipse, hyperbola, and parabola are examples of conic sections. Their quation contains $$x^2 terms, y^2$$ terms, or both. When these terms both appear, are on the same side, and have different coefficients with same signs, the equation is that of an ellipse.
$$\displaystyle{2}{x}{2}-{8}{x}{y}+{3}{y}{2}-{4}={0}$$
Polar equations for conic sections Graph the following conic sections, labeling vertices, foci, directrices, and asymptotes (if they exist). Give the eccentricity of the curve. Use a graphing utility to check your work. $$\displaystyle{r}=\ {\frac{{{10}}}{{{5}\ +\ {2}\ {\cos{\theta}}}}}$$