Geometric Sequences | General Term & Sum of First n Terms

Learning Objectives

By the end of this article, you should be able to:

• Define a geometric sequence and identify its first term \( a \) and common ratio \( r \).
• Find the general term \( T_n \) of a geometric sequence.
• Calculate the sum of the first n terms \( S_n \) (both \( r \neq 1 \) and \( r = 1 \) cases).
• Find the number of terms, common ratio, or first term given other information.
• Solve problems involving geometric means.
• Apply geometric sequences to real-world contexts (compound interest, growth, etc.).

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1. Introduction:

Geometric sequences appear every year in DSE Paper 1 Section A(1) and A(2). Questions may ask you to:

• Find the general term or a specific term of a geometric sequence
• Calculate the sum of the first n terms
• Find the number of terms or the common ratio
• Solve problems involving geometric means or compound interest

These are essential skills that also extend to infinite series and exponential functions.

2. Definition and Key Terms

A geometric sequence (or geometric progression) is a sequence in which the ratio between consecutive terms is constant. This constant ratio is called the common ratio, denoted by \( r \).

• First term: \( a \)
• Common ratio: \( r = \frac{T_2}{T_1} = \frac{T_3}{T_2} = \cdots \), \( r \neq 0 \)

General Term

The nth term of a geometric sequence is given by:

Examples

• \( 3, 6, 12, 24, \ldots \) → \( a = 3, r = 2 \), \( T_n = 3 \cdot 2^{n-1} \)
• \( 81, 27, 9, 3, \ldots \) → \( a = 81, r = \frac{1}{3} \), \( T_n = 81 \cdot \left(\frac{1}{3}\right)^{n-1} = 3^{4-(n-1)} = 3^{5-n} \)

3. Sum of the First n Terms

The sum of the first n terms of a geometric sequence is denoted by \( S_n \). The formula depends on the common ratio:

• If \( r \neq 1 \):
$$ S_n = \frac{a(1 - r^n)}{1 - r} \quad \text{or} \quad \frac{a(r^n - 1)}{r - 1} $$
• If \( r = 1 \), the sequence is constant: \( S_n = na \).

Worked Example

Question: Find the sum of the first 8 terms of the geometric sequence \( 2, 6, 18, 54, \ldots \).

4. Geometric Means

Geometric means are numbers inserted between two given terms so that the resulting sequence is geometric. If \( k \) geometric means are inserted between \( a \) and \( b \), the common ratio is:

Worked Example

Question: Insert two geometric means between 2 and 54.

5. Applications and Word Problems

Geometric sequences are used to model situations with exponential growth or decay. Common examples include:

• Compound interest (bank accounts, loans)
• Population growth (bacteria, animals)
• Depreciation (value of cars, machinery)
• Doubling/halving processes

Worked Example

Question: A sum of $5000 is invested at 6% compound interest per annum. Find the total amount after 4 years.

6. Worked Examples

Example 1: Finding the General Term

Question: The 2nd term of a geometric sequence is 12, and the 5th term is 96. Find the first term and the common ratio.

Example 2: Sum of Terms

Question: Find the sum of the first 10 terms of the geometric sequence \( 1, -2, 4, -8, \ldots \).

Example 3: Geometric Means

Question: Insert 3 geometric means between 5 and 405.

7. DSE-Style Practice Questions

8. Solutions with Explanations

9. Exercise

Click the following link to have
 An Exercise on Geometric Sequences | General Term & Sum of First n Terms

Key Takeaways

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Summary Checklist for Revision

• Identify \( a \) and \( r \)
• General term: \( T_n = a r^{n-1} \)
• Sum formula: \( S_n = \frac{a(1-r^n)}{1-r} \) (for \( r \neq 1 \))
• Special case \( r = 1 \): \( S_n = na \)
• Find \( n \) from \( T_n \) or \( S_n \)
• Geometric means: \( r = \left( \frac{b}{a} \right)^{1/(k+1)} \)
• Applications: compound interest, growth/decay