Geometric Sequences and the Common Ratio in IGCSE Maths

What Is a Geometric Sequence?

A geometric sequence is a list of numbers where each term is found by multiplying the previous term by a fixed number called the common ratio. For example, 2, 6, 18, 54, 162 is a geometric sequence with a common ratio of 3, because each term is three times the previous one.

Unlike arithmetic sequences, where the terms increase or decrease by a constant amount, geometric sequences grow or shrink by a constant multiplier. This means geometric sequences can increase very rapidly or decrease towards zero, depending on the common ratio.

At the IGCSE level, geometric sequences appear in both numerical pattern questions and real-world applications such as compound interest, population growth, and depreciation.

Finding the Common Ratio

The common ratio r is found by dividing any term by the previous term:

r = second term / first term = third term / second term

For the sequence 5, 15, 45, 135: r = 15/5 = 3.

For the sequence 80, 40, 20, 10: r = 40/80 = 0.5.

For the sequence 3, −6, 12, −24: r = −6/3 = −2.

Key observations about the common ratio:

• If r > 1, the sequence increases without bound
• If 0 < r < 1, the sequence decreases towards zero
• If −1 < r < 0, the terms alternate in sign and decrease in magnitude
• If r < −1, the terms alternate in sign and increase in magnitude
• If r = 1, all terms are the same
• If r = 0, the sequence is trivially zero from the second term onwards

The nth Term Formula

The nth term of a geometric sequence is given by:

aₙ = a₁ × r^(n−1)

where a₁ is the first term, r is the common ratio, and n is the position of the term.

Example: Find the 8th term of the geometric sequence with first term 3 and common ratio 2.

a₈ = 3 × 2^(8−1) = 3 × 2⁷ = 3 × 128 = 384.

Example: The 3rd term of a geometric sequence is 12 and the 6th term is 96. Find the common ratio and the first term.

Using the formula: a₃ = a₁ × r² = 12 and a₆ = a₁ × r⁵ = 96.

Dividing: (a₁ × r⁵)/(a₁ × r²) = 96/12, so r³ = 8, giving r = 2.

Substituting back: a₁ × 4 = 12, so a₁ = 3.

Sum of a Geometric Series

When you add the terms of a geometric sequence, you get a geometric series. The sum of the first n terms is:

Sₙ = a₁(1 − rⁿ)/(1 − r) when r ≠ 1

This formula is provided on the IGCSE formula sheet for the Extended tier. Make sure you can identify when to use it and substitute values correctly.

Example: Find the sum of the first 10 terms of the geometric sequence 2, 6, 18, 54, …

Here a₁ = 2, r = 3, n = 10.

S₁₀ = 2(1 − 3¹⁰)/(1 − 3) = 2(1 − 59049)/(−2) = 2(−59048)/(−2) = 59048.

Infinite Geometric Series

When the common ratio satisfies −1 < r < 1, the terms get progressively smaller and the sum approaches a finite limit as the number of terms increases indefinitely. This sum to infinity is:

S∞ = a₁/(1 − r)

This formula applies only when |r| < 1. If |r| ≥ 1, the series diverges and has no finite sum.

Example: Find the sum to infinity of the geometric series 10 + 5 + 2.5 + 1.25 + …

Here a₁ = 10 and r = 0.5.

S∞ = 10/(1 − 0.5) = 10/0.5 = 20.

This means that no matter how many terms you add, the sum will never exceed 20 but gets closer and closer to it.

Real-World Applications

Geometric sequences model many real-world situations:

• Compound interest: An investment of $1000 growing at 5% per year forms a geometric sequence with first term 1000 and common ratio 1.05. After n years, the amount is 1000 × 1.05ⁿ.
• Depreciation: A car worth $20,000 that loses 15% of its value each year has a geometric sequence with first term 20000 and common ratio 0.85.
• Population growth: A bacteria colony that doubles every hour follows a geometric sequence with common ratio 2.
• Radioactive decay: A substance with a half-life follows a geometric sequence with common ratio 0.5.

In exam questions, the challenge is often recognising that a situation involves a geometric sequence and identifying the first term and common ratio from the context.

Common Exam Questions

Cambridge asks geometric sequence questions in several formats:

• Finding the nth term given the first term and common ratio
• Finding the common ratio given two terms
• Setting up and solving equations using the nth term formula
• Finding the sum of a specified number of terms
• Determining whether a sequence converges and finding the sum to infinity
• Real-world problems involving growth or decay

For the most challenging questions, you might need to set up simultaneous equations using two pieces of information about the sequence, solve for a₁ and r, and then find a specific term or sum.

Common Mistakes

Students frequently make these errors:

• Confusing the term number with the power. The nth term uses r^(n−1), not rⁿ. The exponent is always one less than the term number.
• Forgetting that the common ratio can be negative or fractional. Do not assume r is a positive whole number.
• Using the wrong formula. Arithmetic and geometric sequence formulas are very different. Make sure you identify the type of sequence first.
• Sign errors in the sum formula. Be careful with the (1 − r) denominator, especially when r is negative.
• Applying the sum to infinity formula when |r| ≥ 1. The formula only works for convergent series.

Practice Strategy

Start with identifying sequences as geometric and finding the common ratio. Progress to finding nth terms and sums using the formulas. Finally, work on multi-step problems and real-world applications that require you to set up the problem from scratch.

Timing yourself on past paper questions helps build the speed needed for the exam, where geometric sequence questions are usually worth three to five marks.

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