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Change in Arithmetic and Geometric Sequences

Alice White

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 ๐Ÿš€

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? ๐Ÿค”

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. ๐Ÿšถโ€โ™€๏ธ

Graph displaying five points, each point labeled with a different day of the week.


Arithmetic Sequences โž•

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. ๐Ÿ’ก

  • 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 ๐Ÿ˜Ž

Key Concept

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)

Memory Aid

Think: Arithmetic = Adding. You're adding the same difference each time, like climbing stairs with equal steps.

Arithmetic sequence formula an=a+(n-1)d displayed along with an example to find the 46th term of the sequence: 3,8,13,18,23,28,33,โ€ฆ with a = 3 and d = 5. Answer is 228.


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 โœ–๏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. ๐Ÿ’ก

  • 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 ๐Ÿค“

Key Concept

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)

Memory Aid

Think: Geometric = Growing. You're multiplying by the same ratio each time, like compound interest.

Two graphs displayed with one graph having an upward curve with the points plotted and the other graph having a downward curve with the points plotted.


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 ๐Ÿค”

  • Arithmetic: Constant increase (or decrease) by the common difference. Linear growth.
  • Geometric: Increasing rate of increase (or decrease) by the common ratio. Exponential growth.

Memory Aid

Visualize: Arithmetic is like a straight line (constant slope), while geometric is like a curve (increasing slope).

A coordinate plane with three different graphs plotted and the curve upward of them gradually increases. From least curved to most curved is as follows: arithmetic progression, geometric progression, and arithmetic-geometric progression.


Common Mistake

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 ๐ŸŽฏ

  • 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.

Exam Tip

Double-Check: Always double-check your calculations, especially when dealing with exponents in geometric sequences.


Last-Minute Tips ๐Ÿš€

  • 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! ๐Ÿ’ช

Question 1 of 11

Which of the following best describes the graph of a sequence? ๐Ÿค”

A continuous curve

A straight line

Discrete points

A smooth curve