Properties of Geometric Sequences

Besides the basic definition and formulas, geometric sequences have many important properties. Mastering these properties allows you to solve problems more flexibly.

The Middle-Term Property

The Geometric Mean

In other words, the square of the geometric mean equals the product of the two end terms.

Example: $2$, $4$, $8$ form a geometric sequence, and $4$ is the geometric mean of $2$ and $8$, because $4^2 = 2 \times 8 = 16$.

The General Middle-Term Property

In the geometric sequence $\{a_n\}$, for any positive integer $n$ ($n \geq 2$):

This shows that in a geometric sequence, the square of any term equals the product of its two neighbors.

The Product Property

In the geometric sequence $\{a_n\}$, if $m + n = p + q$ (where $m, n, p, q$ are positive integers), then:

In particular, when $m + n = 2p$, we have $a_m \cdot a_n = a_p^2$.

Example: In a geometric sequence, $a_2 \cdot a_6 = a_3 \cdot a_5 = a_4^2$ (because $2+6=3+5=2 \times 4=8$)

Proof:

Since $m + n = p + q$, we have $a_m \cdot a_n = a_p \cdot a_q$.

The Subsequence Property

Extracting at Equal Intervals

Extracting several terms from a geometric sequence at equal intervals produces a new sequence that is still a geometric sequence.

Example: The sequence $\{a_n\}$: $2, 6, 18, 54, 162, \ldots$ (common ratio 3)

Extracting odd-indexed terms: $a_1, a_3, a_5, \ldots$ i.e., $2, 18, 162, \ldots$ (common ratio 9)

Extracting even-indexed terms: $a_2, a_4, a_6, \ldots$ i.e., $6, 54, 486, \ldots$ (common ratio 9)

Products of Consecutive k Terms

Taking the product of every consecutive $k$ terms of a geometric sequence as a term, the new sequence thus formed is still a geometric sequence.

Relationship with the Sum of the First n Terms

The Relationship between the General Term and the Sum of the First n Terms

For a geometric sequence $\{a_n\}$ ($q \neq 1$), the sum $S_n$ of the first $n$ terms and the general term $a_n$ satisfy:

The Property of the Sum of the First n Terms

When $q \neq 1$, the sum of the first $n$ terms of a geometric sequence can be written as:

$S_n = \frac{a_1(1 - q^n)}{1 - q} = \frac{a_1}{1 - q} - \frac{a_1 q^n}{1 - q}$

This shows that $S_n$ can be regarded as the sum of a constant term and an exponential term.

Worked Examples

Example 1: Using the Product Property

In the geometric sequence $\{a_n\}$, given $a_3 \cdot a_5 = 16$, find $a_4$.

Solution:

Since $3 + 5 = 2 \times 4$, by the product property:

$a_3 \cdot a_5 = a_4^2$

Therefore: $a_4^2 = 16 \quad \Rightarrow \quad a_4 = \pm 4$

Example 2: Using the Subsequence Property

The common ratio of the geometric sequence $\{a_n\}$ is $q = 2$. Extract the 1st, 3rd, 5th, … terms to form a new sequence $\{b_n\}$. Find the common ratio of $\{b_n\}$.

Solution:

$b_1 = a_1$, $b_2 = a_3$, $b_3 = a_5$, $\ldots$

$\frac{b_2}{b_1} = \frac{a_3}{a_1} = q^2 = 2^2 = 4$

So the common ratio of the new sequence is $4$.

Example 3: Using the Sum Property

The sum of the first $n$ terms of the geometric sequence $\{a_n\}$ is $S_n = 3^n - 1$. Find $a_1$ and $q$.

Solution:

When $n = 1$: $a_1 = S_1 = 3^1 - 1 = 2$

When $n \geq 2$: $a_n = S_n - S_{n-1} = (3^n - 1) - (3^{n-1} - 1) = 3^n - 3^{n-1} = 3^{n-1}(3 - 1) = 2 \cdot 3^{n-1}$

Verify $n = 1$: $a_1 = 2 \cdot 3^0 = 2$ ✓

So $a_n = 2 \cdot 3^{n-1}$, i.e., $a_1 = 2$ and $q = 3$.

Practice Problems

Exercise 1

In the geometric sequence $\{a_n\}$, $a_2 \cdot a_8 = 36$. Find $a_5$.

Exercise 2

In the geometric sequence $\{a_n\}$, $a_1 \cdot a_2 \cdot a_3 = 27$. Find $a_2$.

Exercise 3

The sum of the first $n$ terms of the geometric sequence $\{a_n\}$ is $S_n = 2^{n+1} - 2$. Find $a_n$.

Exercise 4

Adapted from a postgraduate entrance examination problem

In the geometric sequence $\{a_n\}$, $a_1 > 0$, $q > 1$, $a_1 + a_n = 66$, $a_2 \cdot a_{n-1} = 128$, and $S_n = 126$. Find $n$.

Summary

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