How to solve sequence, angle of a circle, area of a cone median

$a_1 = 3$
$d = 2$
$n = 5$

The 5th term is $\boxed{\textbf{11}}$.

Angle of a Circle

There are several types of angles related to a circle. Two fundamental ones are:

• Central Angle: An angle whose vertex is the center of the circle and whose sides are radii. Its measure is equal to the measure of the intercepted arc.
• Inscribed Angle: An angle whose vertex is on the circle and whose sides are chords of the circle.

Relationship between Central and Inscribed Angles:
An inscribed angle is half the measure of the central angle that subtends the same arc.
If a central angle $\theta_c$ and an inscribed angle $\theta_i$ both intercept the same arc, then:

Example: If a central angle subtends an arc of $80^\circ$, what is the measure of an inscribed angle that subtends the same arc?

$\theta_c = 80^\circ$

The inscribed angle is $\boxed{\textbf{40}^\circ}$.

Area of a Cone

A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (usually circular) to a point called the apex or vertex.

To find the area of a cone, we usually refer to its surface area, which includes the area of its base and its lateral (curved) surface.

Let:

• $r$ be the radius of the circular base
• $h$ be the perpendicular height from the base to the apex
• $l$ be the slant height (the distance from the apex to any point on the circumference of the base)

The formulas are:

• Area of the Base ($A_b$): $$A_b = \pi r^2$$
• Lateral Surface Area ($A_L$): $$A_L = \pi r l$$
• Total Surface Area ($A_T$): $$A_T = A_b + A_L = \pi r^2 + \pi r l = \pi r (r + l)$$

Note: The slant height $l$ can be found using the Pythagorean theorem if $r$ and $h$ are known: $l = \sqrt{r^2 + h^2}$.

Example: Find the total surface area of a cone with a base radius ($r$) of 3 cm and a slant height ($l$) of 5 cm.

$r = 3 \text{ cm}$
$l = 5 \text{ cm}$

The total surface area is $\boxed{\textbf{24}\pi \textbf{ cm}^2}$.

Median

The median is the middle value in a dataset when the data is arranged in numerical order. It represents the central tendency of the data.

How to find the median:

• If $N$ is odd: The median is the middle value. Its position is $\frac{N+1}{2}$.
• If $N$ is even: The median is the average of the two middle values. Their positions are $\frac{N}{2}$ and $\frac{N}{2} + 1$.

Example 1 (Odd number of data points): Find the median of the dataset: 7, 2, 9, 4, 5.

2, 4, 5, 7, 9

$N = 5$

The 3rd value in the ordered list is 5.
The median is $\boxed{\textbf{5}}$.

Example 2 (Even number of data points): Find the median of the dataset: 10, 3, 8, 6.

3, 6, 8, 10

$N = 4$

The 2nd value is 6.
The 3rd value is 8.

The median is $\boxed{\textbf{7}}$.