This worksheet will guide you through the fundamentals of arithmetic and geometric sequences, two crucial concepts in algebra. We'll cover their definitions, formulas, and how to solve common problems. By the end, you'll be comfortable identifying, analyzing, and working with both types of sequences.
What is an Arithmetic Sequence?
An arithmetic sequence is a sequence where the difference between consecutive terms is constant. This constant difference is called the common difference, often denoted by 'd'. Each term is obtained by adding the common difference to the previous term.
Example: 2, 5, 8, 11, 14... (Here, d = 3)
Formula: The nth term of an arithmetic sequence is given by: an = a1 + (n-1)d, where a1 is the first term, n is the term number, and d is the common difference.
What is a Geometric Sequence?
A geometric sequence is a sequence where each term is obtained by multiplying the previous term by a constant value. This constant value is called the common ratio, often denoted by 'r'.
Example: 3, 6, 12, 24, 48... (Here, r = 2)
Formula: The nth term of a geometric sequence is given by: an = a1 * r(n-1), where a1 is the first term, n is the term number, and r is the common ratio.
Identifying Arithmetic and Geometric Sequences
How to tell the difference: The key is to look at the relationship between consecutive terms. If the difference is constant, it's arithmetic. If the ratio is constant, it's geometric.
Example 1: 1, 4, 7, 10, 13... (Arithmetic; d = 3) Example 2: 2, 6, 18, 54, 162... (Geometric; r = 3) Example 3: 1, 2, 4, 7, 11... (Neither arithmetic nor geometric)
Finding the nth Term
Let's practice finding specific terms in arithmetic and geometric sequences.
Problem 1 (Arithmetic): Find the 10th term of the arithmetic sequence 5, 9, 13, 17...
Solution: a1 = 5, d = 4, n = 10. Using the formula an = a1 + (n-1)d, we get a10 = 5 + (10-1)4 = 39.
Problem 2 (Geometric): Find the 7th term of the geometric sequence 2, 6, 18, 54...
Solution: a1 = 2, r = 3, n = 7. Using the formula an = a1 * r(n-1), we get a7 = 2 * 3(7-1) = 1458.
Finding the Common Difference or Ratio
Sometimes, you'll need to find the common difference or ratio given a few terms.
Problem 3: The 3rd term of an arithmetic sequence is 11 and the 7th term is 27. Find the common difference.
Solution: We know that a3 = a1 + 2d = 11 and a7 = a1 + 6d = 27. Subtracting the first equation from the second gives 4d = 16, so d = 4.
Problem 4: The 2nd term of a geometric sequence is 12 and the 5th term is 96. Find the common ratio.
Solution: We have a2 = a1r = 12 and a5 = a1r4 = 96. Dividing the second equation by the first gives r3 = 8, so r = 2.
Applications of Arithmetic and Geometric Sequences
Arithmetic and geometric sequences appear in various real-world applications, including:
- Simple Interest: The yearly balance in a simple interest account forms an arithmetic sequence.
- Compound Interest: The yearly balance in a compound interest account forms a geometric sequence.
- Depreciation: The value of an asset that depreciates by a fixed percentage each year forms a geometric sequence.
- Population Growth (under certain conditions): Population growth can be modeled using geometric sequences.
This worksheet provides a solid foundation for understanding arithmetic and geometric sequences. Remember to practice regularly to master these concepts. Further exploration could include solving more complex problems involving sums of sequences, infinite geometric series, and applications in various fields.
