Series: 2, 6, 12, 20, ?

Unlocking the Mystery of Number Sequences: Solving 2, 6, 12, 20, ?

Have you ever been captivated by the elegance of numbers? Number sequences hold a special fascination, revealing hidden patterns and mathematical truths. Let's delve into the intriguing sequence: 2, 6, 12, 20, ... What comes next? This blog post will solve this mystery and explain the exciting mathematical concepts involved.

Analyzing the Number Sequence

Finding the Differences

To crack this code, let's start by finding the differences between consecutive terms:

• 6 - 2 = 4
• 12 - 6 = 6
• 20 - 12 = 8

Notice something interesting? The differences aren't constant; it's not a simple arithmetic sequence. But let's try something else.

Identifying the Pattern

Let's find the differences between these differences (the second differences):

• 6 - 4 = 2
• 8 - 6 = 2

Aha! The second differences are constant, equal to 2. This means the sequence isn't arithmetic, but rather, it's a quadratic sequence. This reveals a pattern based on a squared term.

Determining the Pattern and the Next Number

Formula Derivation

Since the second differences are constant, the nth term of the sequence can be represented by a quadratic formula of the form: a*n² + b*n + c

Using the first three terms (n=1, 2, 3), we can create a system of equations and solve for a, b, and c. After some algebraic maneuvering (which we'll skip for brevity here), we get the formula: n(n+1)

Let's check:

• For n=1: 1(1+1) = 2
• For n=2: 2(2+1) = 6
• For n=3: 3(3+1) = 12
• For n=4: 4(4+1) = 20

Our formula works!

Calculating the Missing Term

Now, to find the next term (n=5), we substitute n=5 into our formula:

5(5+1) = 30

Therefore, the next number in the sequence is 30.

Conclusion

We've successfully solved the number sequence, determining the missing number to be 30! The solution involved analyzing differences, recognizing a quadratic pattern, deriving a formula, and applying it. This demonstrates the power of pattern recognition in mathematics.

I encourage you to explore other number sequences and challenge yourself to find their underlying patterns. There are many resources available online to further your mathematical journey. Share your thoughts or any other interesting sequences you've encountered in the comments below!