Next In Sequence 2 6 12 20 30 Explained

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Hey guys, ever stumbled upon a sequence that just makes you scratch your head? We've got a fun one here: 2, 6, 12, 20, 30. The challenge is to figure out what number comes next, and we've got some options to choose from: A) 42, B) 56, C) 72, and D) 90. Let's dive into this mathematical puzzle and crack the code together!

Cracking the Code The Logic Behind the Sequence

To figure out the next number in the sequence 2, 6, 12, 20, 30, we first need to identify the pattern. Sequences in mathematics often follow a specific rule or formula, and our job is to uncover that rule. A common approach is to look at the differences between consecutive terms. Let's calculate these differences:

  • The difference between 6 and 2 is 4.
  • The difference between 12 and 6 is 6.
  • The difference between 20 and 12 is 8.
  • The difference between 30 and 20 is 10.

Notice anything interesting? The differences themselves form a sequence: 4, 6, 8, 10. This sequence consists of consecutive even numbers, increasing by 2 each time. This is a crucial clue! It suggests that the original sequence is not a simple arithmetic progression (where the difference between terms is constant) but rather a sequence where the differences increase linearly.

So, if the differences are increasing by 2, the next difference in the sequence 4, 6, 8, 10 should be 12. This means that to find the next number in the original sequence (2, 6, 12, 20, 30), we need to add 12 to the last number, which is 30. Therefore, 30 + 12 = 42. Aha! It seems like option A, 42, is our answer. But let's solidify our understanding and explore a more formal way to represent this sequence.

We can express the sequence using a formula. Since the differences increase linearly, the original sequence is likely a quadratic sequence, meaning it can be represented by a quadratic equation of the form an² + bn + c, where a, b, and c are constants. To find these constants, we can use the first few terms of the sequence. However, for the purpose of this explanation, recognizing the pattern of increasing differences is sufficient to arrive at the answer. The beauty of mathematics lies in the various approaches we can take to solve a problem!

To further confirm our solution, let's think about the nature of quadratic sequences. They arise when there's a constant second difference. In our case, the first differences are 4, 6, 8, 10, and the second differences (the differences between the first differences) are all 2. This confirms that the sequence is indeed quadratic, and our method of finding the next term by adding the next difference is valid. Understanding these underlying principles not only helps us solve this specific problem but also equips us to tackle similar sequence challenges in the future. So, it's not just about getting the right answer; it's about understanding why the answer is right.

The Formulaic Approach Diving Deeper into the Sequence

For those who enjoy a more algebraic approach, let's formulate the sequence mathematically. As we identified, this sequence is quadratic, meaning it follows the form an² + bn + c. To determine the values of a, b, and c, we can use the first three terms of the sequence (2, 6, and 12) and set up a system of equations.

Let's denote the nth term of the sequence as T(n). Then we have:

  • T(1) = a(1)² + b(1) + c = a + b + c = 2
  • T(2) = a(2)² + b(2) + c = 4a + 2b + c = 6
  • T(3) = a(3)² + b(3) + c = 9a + 3b + c = 12

Now we have a system of three equations with three unknowns. We can solve this system using various methods, such as substitution or elimination. Let's use elimination. First, subtract the first equation from the second and the second equation from the third:

  • (4a + 2b + c) - (a + b + c) = 6 - 2 => 3a + b = 4 (Equation 4)
  • (9a + 3b + c) - (4a + 2b + c) = 12 - 6 => 5a + b = 6 (Equation 5)

Now subtract Equation 4 from Equation 5:

  • (5a + b) - (3a + b) = 6 - 4 => 2a = 2 => a = 1

Substitute a = 1 into Equation 4:

  • 3(1) + b = 4 => b = 1

Finally, substitute a = 1 and b = 1 into the first equation:

  • 1 + 1 + c = 2 => c = 0

So, we have a = 1, b = 1, and c = 0. Therefore, the formula for the sequence is T(n) = n² + n. This is a neat way to represent our sequence! Now, to find the next number in the sequence, we need to find the value of T(6) (since we already have the first five terms).

T(6) = 6² + 6 = 36 + 6 = 42

Lo and behold, we arrive at the same answer, 42! This confirms our initial finding and showcases the power of using algebraic methods to solve sequence problems. The formula T(n) = n² + n provides a concise and elegant way to generate any term in the sequence. Understanding the formula not only solves the problem at hand but also provides a deeper insight into the structure of the sequence itself. It's like having a secret code that unlocks the entire sequence!

Choosing the Correct Path A Matter of Perspective

Both approaches we've explored the pattern-based method and the formulaic method are valid and effective for solving this problem. The pattern-based method relies on recognizing the increasing differences and extrapolating from there, while the formulaic method involves finding a mathematical expression that generates the sequence. Which method is