Arithmetic Sequence

An arithmetic sequence is a sequence of terms that have a common difference between them.

General Term: $a_n = a_1 + (n - 1) \cdot d$ where:

• $a_n$ is the $n$th term,
• $a_1$ is the first term,
• $d$ is the common difference.

Geometric Sequence

A geometric sequence is a sequence of terms that have a common ratio between them.

General Term: $a_n = a_1 \cdot r^{(n - 1)}$ where:

• $a_n$ is the $n$th term,
• $a_1$ is the first term,
• $r$ is the common ratio.

Problem 1: Sequence Classification and Identification of $d$ or $r$

Let's determine if the given sequences are arithmetic, geometric, or neither.

a. Sequence: $6, 12, 18, 24, \dots$

• First Term: $a_1 = 6$
• Common Difference: $d = 12 - 6 = 6$

This sequence has a common difference of 6, so it is an arithmetic sequence with $d = 6$.

b. Sequence: $6, 11, 17, \dots$

• First Term: $a_1 = 6$
• Common Difference: $d = 11 - 6 = 5$

This sequence has a common difference of 5, so it is an arithmetic sequence with $d = 5$.

c. Sequence: $2, 14, 98, 686, \dots$

• First Term: $a_1 = 2$
• Common Ratio: $r = \frac{14}{2} = 7$

This sequence has a common ratio of 7, so it is a geometric sequence with $r = 7$.

d. Sequence: $160, 80, 40, 20, \dots$

• First Term: $a_1 = 160$
• Common Ratio: $r = \frac{80}{160} = \frac{1}{2}$

This sequence has a common ratio of $\frac{1}{2}$, so it is a geometric sequence with $r = \frac{1}{2}$.

e. Sequence: $-40, -25, -10, 5, \dots$

• First Term: $a_1 = -40$
• Common Difference: $d = -25 - (-40) = 15$

This sequence has a common difference of 15, so it is an arithmetic sequence with $d = 15$.

f. Sequence: $7, -21, 63, -189, \dots$

• First Term: $a_1 = 7$
• Common Ratio: $r = \frac{-21}{7} = -3$

This sequence has a common ratio of $-3$, so it is a geometric sequence with $r = -3$.

Summary

• a. Arithmetic, $d = 6$
• b. Arithmetic, $d = 5$
• c. Geometric, $r = 7$
• d. Geometric, $r = \frac{1}{2}$
• e. Arithmetic, $d = 15$
• f. Geometric, $r = -3$

Do you need more details or have any questions?

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1. What is the general term formula for an arithmetic sequence?
2. How do you find the common ratio in a geometric sequence?
3. What is the next term in the sequence $6, 12, 18, 24, \dots$?
4. Identify the type of sequence and find the 5th term of the sequence $2, 14, 98, 686, \dots$.
5. Can a sequence be both arithmetic and geometric? Explain why or why not.

Tip: When determining if a sequence is arithmetic, subtract consecutive terms to check for a constant difference. For geometric sequences, divide consecutive terms to check for a constant ratio.