Sequences and Series

Furthermore

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio, denoted by \( r \).

The \( n^{th} \) term of a geometric sequence can be calculated using the formula:

$$ u_n = u_1 \cdot r^{(n - 1)} \\ \text{where } u_n \text{ is the } n^{th} \text{ term, } u_1 \text{ is the first term, and } r \text{ is the common ratio.} $$

The sum of the first \( n \) terms (\( S_n \)) of a geometric sequence can be found using the formula:

$$ S_n = \frac{u_1(1 - r^n)}{1 - r} \text{, for } r \neq 1 \\ \text{where } S_n \text{ is the sum of the first } n \text{ terms.} $$

Sigma notation (\( \Sigma \)) is also used to represent the sum of geometric sequences. For a geometric sequence, it can be represented as:

$$ S_n = \sum_{k=0}^{n-1} u_1 \cdot r^k $$

Example 1: Consider a geometric sequence where the first term \( u_1 = 2 \) and the common ratio \( r = 3 \). Let's find the \( 5^{th} \) term and the sum of the first 5 terms.

Using the formula for the \( n^{th} \) term:

$$ u_5 = 2 \cdot 3^{(5 - 1)} = 2 \cdot 81 = 162 $$

So, the \( 5^{th} \) term is 162.

Now, using the formula for the sum of the first \( n \) terms:

$$ S_5 = \frac{2(1 - 3^5)}{1 - 3} = \frac{2(1 - 243)}{-2} = \frac{484}{2} = 242 $$

Thus, the sum of the first 5 terms of the given geometric sequence is 242.

Example 2: For a geometric sequence with the first term \( u_1 = 4 \) and the common ratio \( r = 0.5 \), we can represent the sum of the first 4 terms using sigma notation as:

$$ S_4 = \sum_{k=0}^{3} 4 \cdot (0.5)^k $$

This notation concisely represents the sum of the sequence: 4, 2, 1, and 0.5, which equals 7.5.

This video on Geometric Sequences is from Revision Village and is aimed at students taking the IB Maths Standard level course.