Word Problems On Geometric Sequence

Decoding the Mystery: Mastering Word Problems on Geometric Sequences

Geometric sequences, those elegant patterns where each term is a constant multiple of the previous one, often hide within seemingly complex word problems. Understanding these problems isn't just about plugging numbers into formulas; it's about translating real-world scenarios into mathematical language and applying the power of geometric sequences to find solutions. This comprehensive guide will equip you with the tools and strategies to confidently tackle even the trickiest word problems involving geometric sequences. We'll cover fundamental concepts, step-by-step problem-solving techniques, explore diverse examples, and delve into the underlying mathematical principles.

Understanding the Fundamentals of Geometric Sequences

Before diving into word problems, let's solidify our understanding of geometric sequences. A geometric sequence is a sequence where each term is obtained by multiplying the previous term by a constant value, called the common ratio (often denoted by 'r'). The general formula for the nth term of a geometric sequence is:

a_n = a₁ * r^(n-1)

Where:

• a_n is the nth term in the sequence.
• a₁ is the first term in the sequence.
• r is the common ratio.
• n is the term number.

The sum of the first n terms of a geometric sequence (also known as a geometric series) is given by:

S_n = a₁ * (1 - rⁿ) / (1 - r) (where r ≠ 1)

Understanding these formulas is crucial, but equally important is the ability to identify a geometric sequence within a word problem's context. Look for scenarios describing repeated multiplication or exponential growth/decay.

Step-by-Step Approach to Solving Geometric Sequence Word Problems

Solving word problems requires a systematic approach. Here's a step-by-step method that will guide you through the process:

1. Identify the Problem Type: Carefully read the problem statement to determine if it involves a geometric sequence. Look for keywords like "doubles," "triples," "increases by a factor of," "decays exponentially," or similar phrases indicating a constant multiplicative relationship between successive terms.

2. Define Variables: Assign variables to the unknown quantities. This often includes the first term (a₁), the common ratio (r), the number of terms (n), and the value of a specific term (a_n) or the sum of terms (S_n).

3. Translate the Problem into Mathematical Equations: Express the given information and the unknown quantities using the formulas for geometric sequences (a_n and S_n). This involves translating the word problem's description into mathematical equations.

4. Solve the Equations: Use algebraic manipulation to solve for the unknown variables. This may involve using logarithms if the exponent (n) is unknown.

5. Check Your Answer: Always check if your answer makes sense in the context of the problem. Does the answer seem reasonable given the scenario described?

Diverse Examples and Solutions

Let's work through several examples to illustrate the application of this methodology.

Example 1: The Bouncing Ball

A ball is dropped from a height of 10 meters. Each time it bounces, it reaches 70% of its previous height. What height does the ball reach after the third bounce?

Solution:

1. Problem Type: This is a geometric sequence problem because the height after each bounce is a constant fraction (70%) of the previous height.

2. Variables:

   • a₁ = 10 meters (initial height)
   • r = 0.7 (common ratio)
   • n = 4 (4th term represents the height after the 3rd bounce)
   • a_n = ? (height after the third bounce)

3. Equation: We use the formula a_n = a₁ * r^(n-1)

4. Solve: a₄ = 10 * (0.7)^(4-1) = 10 * (0.7)³ = 3.43 meters

5. Check: The answer (3.43 meters) is reasonable since the height decreases with each bounce.

Example 2: Bacterial Growth

A bacterial culture starts with 100 bacteria and doubles every hour. How many bacteria will there be after 5 hours?

Solution:

1. Problem Type: Exponential growth indicates a geometric sequence.

2. Variables:

   • a₁ = 100
   • r = 2
   • n = 6 (6th term represents the population after 5 hours)
   • a_n = ?

3. Equation: a_n = a₁ * r^(n-1)

4. Solve: a₆ = 100 * 2^(6-1) = 100 * 2⁵ = 3200 bacteria

5. Check: The exponential growth is reflected in the significantly larger population after 5 hours.

Example 3: Compound Interest

An investment of $5000 earns 5% interest compounded annually. What is the value of the investment after 3 years?

Solution:

1. Problem Type: Compound interest is a classic example of a geometric sequence.

2. Variables:

   • a₁ = 5000
   • r = 1.05 (1 + 0.05)
   • n = 4 (4th term represents the value after 3 years)
   • a_n = ?

3. Equation: a_n = a₁ * r^(n-1)

4. Solve: a₄ = 5000 * (1.05)^(4-1) = 5000 * (1.05)³ ≈ $5788.13

5. Check: The final value is greater than the initial investment, reflecting the accumulation of interest.

Example 4: Depreciation

A car depreciates at a rate of 15% per year. If the initial value was $25,000, what is its value after 4 years?

Solution:

1. Problem Type: Depreciation is an example of exponential decay, representing a geometric sequence.

2. Variables:

   • a₁ = 25000
   • r = 0.85 (1 - 0.15)
   • n = 5 (5th term represents the value after 4 years)
   • a_n = ?

3. Equation: a_n = a₁ * r^(n-1)

4. Solve: a₅ = 25000 * (0.85)^(5-1) = 25000 * (0.85)⁴ ≈ $13,050.64

5. Check: The value decreases over time, reflecting the depreciation.

Addressing Common Challenges and FAQs

Q1: How do I handle problems where the common ratio is not explicitly stated?

A1: Often, the problem will provide information allowing you to calculate the common ratio. Look for relationships between consecutive terms. For example, if you know the second and first terms, you can find 'r' by dividing the second term by the first (r = a₂ / a₁).

Q2: What if the problem asks for the number of terms (n)?

A2: This requires using logarithms. If you know a_n, a₁, and r, you can solve for n using the formula a_n = a₁ * r^(n-1). Take the logarithm of both sides to isolate (n-1) and then solve for n.

Q3: How do I approach problems involving the sum of a geometric series?

A3: Use the formula for the sum of a geometric series (S_n = a₁ * (1 - rⁿ) / (1 - r)). Identify the known variables (a₁, r, n) and solve for the unknown (S_n) or vice versa.

Q4: What if the problem involves a scenario with both growth and decay phases?

A4: Break the problem into smaller segments, applying the appropriate geometric sequence formula to each phase. Then combine the results to answer the overall question.

Conclusion: Unlocking the Power of Geometric Sequences

Word problems on geometric sequences provide a powerful application of mathematical concepts to real-world situations. By understanding the fundamental formulas, adopting a systematic approach, and practicing with diverse examples, you can develop confidence and proficiency in solving these seemingly complex problems. Remember that the key lies in carefully translating the word problem's description into mathematical language and then applying the relevant formulas effectively. With practice and persistence, you will master the art of decoding the mysteries hidden within these intriguing mathematical puzzles.