Algebra 2 - Sequences & Series Flashcards

Questions and Answers

What is a sequence?

A collection of numbers that follow a particular pattern.

What is an arithmetic sequence?

A sequence with a common difference between two consecutive numbers constant.

What is a geometric sequence?

A sequence with a common ratio between two consecutive numbers constant.

What is the common difference in an arithmetic sequence?

The constant number that is added, which is the difference between two numbers. Signup and view all the answers

What is the common ratio in a geometric sequence?

The constant number that is multiplied, which is the ratio between two numbers. Signup and view all the answers

What is the formula for an arithmetic sequence?

an = a1 + d(n - 1) Signup and view all the answers

How do you find the 4th term in the arithmetic sequence 3, 6, 9,...?

a4 = 3 + 2(4 - 1) = 12 Signup and view all the answers

What is the formula for a geometric sequence?

an = a1(r)^(n - 1) Signup and view all the answers

How would you find the 10th term in the geometric sequence 2, 4, 8, 16,...?

a10 = 2(2)^(10 - 1) = 1024 Signup and view all the answers

What is an arithmetic series?

The sum of various numbers in an arithmetic sequence. Signup and view all the answers

What is the formula for an arithmetic series?

Sn = n/2(t1 + tn) Signup and view all the answers

What is a geometric series?

The sum of various numbers in a geometric sequence. Signup and view all the answers

When is a geometric series considered infinite?

When |r| < 1. Signup and view all the answers

What does Sigma(Σ) represent?

It means to sum up and is used to indicate a series. Signup and view all the answers

How many terms are in the sequence 18, 22, 26, 30,..., 110?

24 Signup and view all the answers

How many multiples of 5 are between 300 and 1000?

139 Signup and view all the answers

What is the common difference of the sequence 13, 10.9, 8.8, 6.7,...?

2.1 Signup and view all the answers

What is the common ratio of the sequence 3.1, 9.3, 27.9,...?

3 Signup and view all the answers

How many rows and bricks are in the pile with 85 at the bottom and 1 at the top?

15 rows, 645 bricks Signup and view all the answers

What is the sum of the first 6 terms in the geometric sequence 5, 10, 20,...?

315 Signup and view all the answers

What is the sum of the first 25 even integers?

650 Signup and view all the answers

When will Josh's and Sophia's money sum up to 200?

In 1 month. Signup and view all the answers

What is the sum of the series 100 + 25 + 25/4 + 25/8 +...?

200 Signup and view all the answers

How far will a rubber ball travel before coming to rest if dropped from 729 cm?

3645 cm Signup and view all the answers

Study Notes

Sequences

A sequence is a collection of numbers arranged in a specific pattern, such as 2, 4, 8, 16, 32, which follows a doubling pattern.

Arithmetic Sequences

Defined by a common difference between consecutive terms, e.g., 3, 6, 9, where the difference is consistently 3.
The formula for finding the nth term is ( a_n = a_1 + d(n-1) ).
An arithmetic series is the sum of terms and can be calculated using ( S_n = \frac{n}{2}(t_1 + t_n) ).

Geometric Sequences

Characterized by a constant ratio between consecutive terms, like in 3, 9, 27, where the ratio is 3.
The nth term can be calculated with the formula ( a_n = a_1(r)^{n-1} ).
Geometric series are sums of terms, with finite series represented as ( S_n = t_1\frac{(1-r^n)}{(1-r)} ) and infinite series (|r| < 1) as ( S_{\infty} = \frac{t_1}{(1-r)} ).

Common Differences and Ratios

Common difference refers to the value added in an arithmetic sequence, while common ratio is the multiplied value in a geometric sequence.

Recursive Formulas

Recursive formulas for sequences involve using the previous term to determine the next term. For arithmetic: ( t_n = t_{n-1} + d ) and for geometric: ( t_n = t_{n-1}(r) ).

Sigma Notation

Sigma (Σ) represents summation, allowing for easier expression of series.

Problem-Solving Techniques

Use arithmetic sequence formulas to determine unknowns, such as the number of terms in a sequence. For example, finding terms between 18, 22, 26, and 110 involves reversing the arithmetic formula.
Count multiples or terms by constructing arithmetic sequences and determining n terms using appropriate formulas.

Practice Examples

Summing series can be practiced by applying the relevant formulas to determine the total sum of sequences or series, such as finding sums of the first 25 even integers or the total height traveled by a dropping ball.

Real-Life Applications

Understanding sequences and series can help in financial planning, predicting trends in data, and calculating quantities in everyday scenarios.

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Description

Test your knowledge of sequences and series in Algebra 2 with these flashcards. Each card provides a definition and an example to enhance your understanding of important concepts like arithmetic and geometric sequences.