Unit 2: Geometric Sequences

A geometric sequence is a sequence of numbers where each term after the first is obtained by multiplying the previous term by a constant ratio, r.

It is expressed in the general form:

a, ar, ar², ar³, ar⁴, ..., ar^n-1

Where:

• a is the first term.
• r is the common ratio.
• n is the term number.
• T_n = ar^n-1 is the formula for the nth term.

1. Common Ratio (r): The ratio between consecutive terms is always the same and is found by:

   r = T_n+1 / T_n

2. Formula for the nth Term:

   T_n = ar^n-1

3. Sum of the First n Terms:

   S_n = a(1 - rⁿ) / (1 - r)

   Where r ≠ 1.

4. Infinite Sum (for |r| < 1):

   S_∞ = a / (1 - r)

Example 1

Problem: Given the sequence 2, 6, 18, 54, ..., find the 5th term.

Solution:

• First term, a = 2
• Common ratio, r = 6 / 2 = 3
• Term number, n = 5

Using the formula:

T₅ = 2 × 3^5-1 = 2 × 3⁴ = 2 × 81 = 162

Therefore, the 5th term is 162.

Example 2

Problem: Given T₁ = 81 and r = 1/3, find the 6th term.

Solution:

Using the formula:

T₆ = 81 × (1/3)^6-1 = 81 × (1/3)⁵

Calculating:

T₆ = 81 × 1/243 = 81 / 243 = 1/3

Therefore, the 6th term is 1/3.

1. Find the 8th term of the geometric sequence 3, 6, 12, 24, ...
2. If the first term is 5 and the common ratio is 2, find the 7th term.
3. Determine if the sequence 4, 12, 36, 108, ... is geometric. If so, find the 10th term.
4. The 3rd term is 16 and the 6th term is 128. Find the first term and the common ratio.

1. What is the common ratio in the sequence 2, 10, 50, 250, ...?

2

5

10

25

2. In the formula T_n = ar^n-1, what does 'r' represent?

The first term

The common ratio

The number of terms

The nth term

Submit Quiz Show Hint

Hint: The common ratio is the factor by which you multiply to get from one term to the next!

• A geometric sequence has a constant ratio between consecutive terms.
• The nth term formula is essential for finding any term in the sequence.
• Geometric sequences have applications in various real-world scenarios, including finance and population growth.