Arithmetic and Geometric Series Review

Comprehensive flashcards covering the definitions, visualizations, and real-world applications of arithmetic and geometric series.

What is an Arithmetic Sequence?

A list of numbers with a common difference ($d$), such as $4, 8, 12, 16$ where $d = 4$.

What is a Geometric Sequence?

A list of numbers with a common ratio ($r$), such as $3, 6, 12, 24$ where $r = 2$.

How is summation defined in the context of sequences?

The process of adding the terms of a sequence together.

What is an Arithmetic Series?

The indicated sum of the terms of an arithmetic sequence, representing constant growth over time.

How do you transition an arithmetic sequence into an arithmetic series?

By replacing the commas in the sequence with plus signs (e.g., $10, 20, 30, 40$ becomes $10 + 20 + 30 + 40$).

How is the distance between jumps described when visualizing an arithmetic series on a number line?

The distance of each jump increases at a steady, linear rate.

What characterizes a Geometric Series?

It is the sum of the terms of a geometric sequence where each term is multiplied by a non-zero constant common ratio ($r$).

What does a geometric series represent in terms of growth?

Exponential growth or decay.

What real-world analogy is used to visualize geometric growth on a number line?

A viral infection simulation where $1$ person infects $3$ others.

How can you test if a series is arithmetic or geometric?

Subtract terms to check for a constant difference ($d$) for arithmetic, or divide terms to check for a constant ratio ($r$) for geometric.

Why is Simple Interest classified as an Arithmetic Series?

Because the interest amount added each period is constant.

Why is Compound Interest classified as a Geometric Series?

Because interest is earned on both the principal and the accumulated interest.

According to the notes, what is considered the 'eighth wonder of the world'?

Wealth building through geometric series (compounding).

What technology tip is provided for calculating the sum of a series in spreadsheets?

Input the "=SUM" formula.

What is a 'Limit' in the context of infinite geometric series?

A value that a sum approaches but never exceeds.

What example is provided for an infinite series with a finite sum?

A bouncing ball reaching $\frac{1}{2}$ its height each time.

For the series $100 + 150 + 200 + 250$, what is the common difference ($d$)?

$$d = 50$$

For the series $1 + 4 + 16 + 64$, what is the common ratio ($r$)?

$$r = 4$$

Is a population doubling every year an example of an arithmetic or geometric series?

Geometric series

What is the name of the constant value multiplied by each term in a geometric series?

Common ratio ($r$)

What are the first three terms of an arithmetic series where $a_1 = 5$ and $d = 10$?

$$5 + 15 + 25$$