Geometric Progression Calculator

What is a Geometric Sequence?

Imagine you're folding a piece of paper in half, again and again. Each time you fold, the paper gets twice as thick. This is like a geometric sequence! In a geometric sequence, each number is found by multiplying the previous number by a fixed amount, called the common ratio.

How to Calculate a Geometric Sequence

To find numbers in a geometric sequence, we start with the first number and keep multiplying by the common ratio. It's like a multiplication train, where each new car is bigger (or smaller) by the same factor!

Formula

The formula for the nth term of a geometric sequence is:

\[ a_n = a_1 \cdot r^{n-1} \]

Where:

• \(a_n\) is the nth term in the sequence
• \(a_1\) is the first term
• \(r\) is the common ratio
• \(n\) is the position of the term we want to find

Calculation Steps

1. Start with the first term (\(a_1\))
2. Multiply by the common ratio (\(r\)) to get the second term
3. Keep multiplying by \(r\) for each new term
4. To find any term directly, use the formula \(a_n = a_1 \cdot r^{n-1}\)

Example

Let's make a geometric sequence with \(a_1 = 2\) and \(r = 3\):

• First term: \(a_1 = 2\)
• Second term: \(a_2 = 2 \cdot 3 = 6\)
• Third term: \(a_3 = 6 \cdot 3 = 18\)
• Fourth term: \(a_4 = 18 \cdot 3 = 54\)

Visual Representation