Geometric Sequences: A Comprehensive GCSE Mathematics Guide
Introduction
- What is a Geometric Sequence?
A geometric sequence is a series of numbers where each term is obtained by multiplying the previous term by a constant called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric sequence with a common ratio of 3.
- Importance in GCSE Mathematics
Geometric sequences are essential in GCSE Mathematics for problem-solving in areas such as finance, growth and decay models, and probability.
- RealWorld Applications
- Compound interest on savings
- Population growth and spread of diseases
- Halflife of radioactive elements
Main Content
Key Concepts and Definitions
- First term (a): The value of the first term in the sequence.
- Common ratio (r): The multiplier that takes one term to the next.
- Nth term (an): The value of the nth term in the sequence.
Formula for the Nth Term
```
an = a1 * r^(n-1)
```
where:
- an is the nth term
- a1 is the first term
- r is the common ratio
- n is the position of the term
Example 1: Finding the Nth Term
Find the 10th term of the geometric sequence 2, 6, 18, 54, ...
- First term (a1) = 2
- Common ratio (r) = 3
- Position of term (n) = 10
```
a10 = 2 * 3^(10-1) = 2 * 3^9 = 364,320
```
Step-by-Step Explanation
1. Identify the first term (a1) as 2.
2. Determine the common ratio (r) as 3.
3. Substitute a1, r, and n into the formula an = a1 * r^(n-1).
4. Calculate a10 = 364,320.
Common Mistakes to Avoid
- Mixing up the first term and the common ratio.
- Forgetting to subtract 1 from n when using the formula.
- Assuming that all sequences are geometric sequences.
Practice Problems
1. Find the 5th term of the sequence 4, 8, 16, 32, ...
2. Write the first 6 terms of the geometric sequence with a1 = 10 and r = 1/2.
3. A bacteria population doubles every hour. If there are initially 500 bacteria, how many bacteria will there be after 6 hours?
Conclusion
Geometric sequences are an important concept in GCSE Mathematics. By understanding the key concepts and formulas, students can confidently solve problems and succeed in their exams.
- Tips for Exam Success
- Memorize the formula for the nth term.
- Practice finding the first term and common ratio.
- Check your answers carefully for any errors.
FAQ
- What is the difference between an arithmetic and geometric sequence?
In an arithmetic sequence, the difference between terms is constant, while in a geometric sequence, the ratio of terms is constant.
- How can I recognize a geometric sequence?
Look for a constant ratio between consecutive terms.