$$\DeclareMathOperator{cosec}{cosec}$$

# 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.

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