Change in Arithmetic and Geometric Sequences

Alice White
6 min read
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Study Guide Overview
This study guide covers arithmetic and geometric sequences. Key concepts include defining sequences, identifying the common difference (d) in arithmetic sequences and the common ratio (r) in geometric sequences. It also provides formulas for finding the nth term of both types of sequences and explains the difference between change and rate of change. Practice questions are included.
AP Pre-Calculus: Sequences - Your Night-Before Review ๐
Hey there! Let's get you prepped for your AP Pre-Calculus exam with a quick, focused review of sequences. We'll break down arithmetic and geometric sequences, highlighting key formulas and concepts to make sure you're feeling confident. Let's do this!
2.1 Change in Arithmetic and Geometric Sequences
What is a Sequence? ๐ค
A sequence is essentially a function that maps whole numbers to real numbers. Think of it as an ordered list of numbers, where each number has a specific position. The graph of a sequence consists of discrete points, not a continuous curve, because we're only dealing with whole numbers as inputs.
For example, if you track your daily steps, each day (1st, 2nd, 3rd, etc.) corresponds to a specific number of steps. These points are plotted separately, not connected by a line. ๐ถโโ๏ธ
Arithmetic Sequences โ
An arithmetic sequence is a sequence where the difference between any two consecutive terms is constant. This constant difference is called the common difference, denoted by d. It's like adding the same number each time to get the next term. ๐ก
- Constant Rate of Change: The terms in an arithmetic sequence increase or decrease at a constant rate, which is equal to the common difference.
- Example: 2, 5, 8, 11, 14... (Here, d = 3).
Formula and Example ๐
The nth term of an arithmetic sequence can be found using these formulas:
- a_n = a_0 + dn (where a_0 is the initial term)
- a_n = a_k + d(n-k) (where a_k is any known term)
Think: Arithmetic = Adding. You're adding the same difference each time, like climbing stairs with equal steps.
Example: Find the 10th term of the arithmetic sequence 4, 7, 10, ...
- a_0 = 4, d = 3
- a_10 = 4 + 3(10) = 34
Geometric Sequences โ๏ธ
A geometric sequence is a sequence where the ratio between any two consecutive terms is constant. This constant ratio is called the common ratio, denoted by r. It's like multiplying by the same number each time to get the next term. ๐ก
- Constant Proportional Change: The terms in a geometric sequence change by a constant proportion, which is equal to the common ratio.
- Example: 2, 6, 18, 54, ... (Here, r = 3).
Formula ๐ค
The nth term of a geometric sequence can be found using these formulas:
- g_n = g_0 * r^n (where g_0 is the initial term)
- g_n = g_k * r^(n-k) (where g_k is any known term)
Think: Geometric = Growing. You're multiplying by the same ratio each time, like compound interest.
Example: Find the 7th term of the geometric sequence 3, 6, 12, ...
- g_0 = 3, r = 2
- g_7 = 3 * 2^7 = 384
Change vs. Rate of Change ๐ค
- Arithmetic: Constant increase (or decrease) by the common difference. Linear growth.
- Geometric: Increasing rate of increase (or decrease) by the common ratio. Exponential growth.
Visualize: Arithmetic is like a straight line (constant slope), while geometric is like a curve (increasing slope).
Don't mix them up! Arithmetic sequences have a common difference (addition/subtraction), while geometric sequences have a common ratio (multiplication/division).
Final Exam Focus ๐ฏ
- High-Priority Topics: Arithmetic and geometric sequences, their formulas, and how to apply them. Also, be comfortable with identifying the common difference or ratio.
- Common Question Types: Finding a specific term, determining if a sequence is arithmetic or geometric, and using the formulas to solve problems.
- Time Management: Quickly identify the type of sequence and use the appropriate formula. Practice will help you recognize these patterns faster.
Double-Check: Always double-check your calculations, especially when dealing with exponents in geometric sequences.
Last-Minute Tips ๐
- Stay Calm: You've got this! Take deep breaths and approach each problem step-by-step.
- Read Carefully: Pay close attention to what the question is asking. Don't jump to conclusions.
- Show Your Work: Even if you make a mistake, you might get partial credit for showing your work.
Practice Questions
Practice Question
Multiple Choice Questions
Question 1:
Which of the following sequences is arithmetic?
(A) 1, 2, 4, 8, ... (B) 2, 5, 8, 11, ... (C) 1, 4, 9, 16, ... (D) 3, 6, 12, 24, ...
Answer: (B)
Question 2:
What is the 8th term of the geometric sequence 5, 10, 20, ...?
(A) 320 (B) 640 (C) 1280 (D) 2560
Answer: (B)
Question 3:
The 4th term of an arithmetic sequence is 18, and the 7th term is 30. What is the common difference?
(A) 2 (B) 3 (C) 4 (D) 6
Answer: (C)
Free Response Question
Question:
A geometric sequence has a first term of 3 and a common ratio of 2. (a) Write the first four terms of the sequence. (b) Find the 10th term of the sequence. (c) If the kth term of the sequence is 1536, find the value of k.
Answer:
(a) The first four terms are: 3, 6, 12, 24 (1 point)
(b) The 10th term is g_10 = 3 * 2^10 = 3 * 1024 = 3072 (2 points)
(c) To find k, we have 3 * 2^(k-1) = 1536. Dividing by 3, we get 2^(k-1) = 512. Since 2^9 = 512, we have k-1 = 9, so k = 10 (2 points)
Good luck on your exam! You've got this! ๐ช

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Question 1 of 11
Which of the following best describes the graph of a sequence? ๐ค
A continuous curve
A straight line
Discrete points
A smooth curve