Imagine starting with a single grain of rice on the first square of a chessboard. On the second square, you place two grains, four on the third, eight on the fourth, and so on, doubling the amount each time. How much rice would you have by the time you reach the 64th square? The answer is an unimaginably colossal number – more rice than has ever been produced in the history of the world!
This classic story illustrates the power of a Geometric Sequence. Unlike arithmetic sequences where you add a constant difference, in geometric sequences, you multiply by a constant factor. And as you’ll see, this can lead to some truly explosive growth (or rapid shrinking!). Welcome back to Sequentia, where today we explore the fascinating world of geometric progressions.
What is a Geometric Sequence? The Core Idea
A geometric sequence (or geometric progression) is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio (often denoted by ‘r’).
Formula:
The nth term of a geometric sequence can be found using the formula:
a_n = a_1 * r^(n-1)
Where:
- a_n is the nth term
- a_1 is the first term
- r is the common ratio
- n is the term number
Examples to Get Started:
- Sequence: 2, 4, 8, 16, 32, …
- First term (a_1): 2
- Common ratio (r): 2 (each term is multiplied by 2 to get the next)
- Sequence: 81, 27, 9, 3, 1, 1/3, …
- First term (a_1): 81
- Common ratio (r): 1/3 (each term is multiplied by 1/3, so it’s shrinking)
- Sequence: 5, -10, 20, -40, 80, …
- First term (a_1): 5
- Common ratio (r): -2 (the sign alternates because the ratio is negative)
Fun Examples of Geometric Growth (and Decay!):
- The Bouncing Ball: Imagine dropping a super bouncy ball. Each time it hits the ground, it bounces back up to, say, 80% of its previous height. The heights of successive bounces (100cm, 80cm, 64cm, 51.2cm…) form a geometric sequence with a common ratio of 0.8. Here, the sequence is decreasing.
(AI Image Prompt for “Bouncing Ball”: See specific idea below) - Compound Interest (Simplified): If you invest money and it earns, for example, 5% interest per year (and you reinvest the interest), the total amount of money you have each year grows geometrically. Your initial amount is multiplied by 1.05 each year. (This is a simplified view ignoring more complex compounding, but illustrates the principle).
- Viral Spread (Ideas or Content): Think about how a funny video or a great idea can spread. One person tells two friends, each of those tells two more friends, and so on. If everyone shares successfully, the number of people who’ve heard it can grow as 1, 2, 4, 8, 16… a geometric sequence!
- Folding Paper: Can you fold a standard piece of paper in half more than 7 or 8 times? Each fold doubles the thickness. The thickness forms a geometric sequence (1 layer, 2 layers, 4 layers, 8 layers…). Very quickly, the paper becomes too thick and small to fold!
Geometric Sequences in Puzzles:
Just like their arithmetic cousins, geometric sequences are a staple in puzzles:
- Identifying the Ratio: The core task is often to figure out the common ratio ‘r’ and then predict the next term.
- Fractional or Negative Ratios: Puzzles can be trickier if ‘r’ is a fraction (leading to shrinking numbers) or negative (leading to alternating signs).
- Missing Terms: Sometimes you’re given a few terms with gaps and need to deduce the ratio to fill them in.
The key is to look for that consistent multiplicative relationship between terms. If dividing a term by its preceding term always gives you the same number, you’ve likely found your common ratio!
From ancient chessboard problems to understanding how investments grow or ideas spread, geometric sequences are a powerful mathematical tool and a fun concept to master for any puzzle enthusiast. Can you think of other real-world examples? Share them below!
