Geometric Sequences Explained with Simple Animations

Welcome to geometric sequences! Today we'll explore patterns where numbers grow or shrink by multiplication.Let's look at the sequence two, six, eighteen, fifty-four.Notice how each number is multiplied by three to get the next term. This number three is called the common ratio.We can visualize this pattern using shapes. Watch how each square grows by a factor of three.The key to a geometric sequence is that we multiply by the same number, called the common ratio, each time.In our example, we multiply by three each time. Two times three gives us six, six times three gives us eighteen, and eighteen times three gives us fifty-four.Let's break down each multiplication step to see how the sequence grows.To find the common ratio in a geometric sequence, we divide each term by the previous term.Let's calculate the ratio between consecutive terms.The nth term formula allows us to find any term in the sequence without calculating all previous terms.Let's understand what each part of the formula means.Let's use this formula to find the fifth term in our sequence.Starting with our first term of 2 and common ratio of 3, we substitute n equals 5 into our formula.Notice how each term is three times larger than the previous term, following our common ratio.Let's explore how bacteria growth follows a geometric sequence. Starting with 100 bacteria that double every hour.Compound interest is another perfect example of geometric sequences. Let's see how $1000 grows at 10% annual interest.Population growth also follows geometric sequences. Let's visualize a population growing by 50% each generation.