Exploring the Sequence 1 7 19 37 61 and Its Quadratic Pattern
The sequence 1, 7, 19, 37, 61 has an interesting pattern that can be unraveled step-by-step. This article will guide you through the reasoning and the mathematical derivation to find the next term in the sequence.
Understanding the Sequence
The given sequence is 1, 7, 19, 37, 61. To find the next term, we can look for patterns in the differences between consecutive terms. Let's start by calculating the differences:
1st differences: 7 - 1 6, 19 - 7 12, 37 - 19 18, 61 - 37 24Next, we examine the differences in these first differences:
2nd differences: 12 - 6 6, 18 - 12 6, 24 - 18 6Since the second differences are constant, it indicates that the original sequence is quadratic. The general form for the nth term of a quadratic sequence is given by:
an An2 Bn C
Deriving the Quadratic Formula
To find the values of A, B, and C, we use the first few terms of the sequence:
n 1: A(1)2 B(1) C 1 rarr; A B C 1 n 2: A(2)2 B(2) C 7 rarr; 4A 2B C 7 n 3: A(3)2 B(3) C 19 rarr; 9A 3B C 19We now have a system of equations to solve:
1. A B C 1
2. 4A 2B C 7
3. 9A 3B C 19
Solving the Equations
First, we subtract equation 1 from equation 2:
(4A 2B C) - (A B C) 7 - 1 rarr; 3A B 6
So, B 6 - 3A.
Next, subtract equation 1 from equation 3:
(9A 3B C) - (A B C) 19 - 1 rarr; 8A 2B 18 rarr; 4A B 9
Substitute B 6 - 3A into 4A B 9:
4A (6 - 3A) 9 rarr; 4A 6 - 3A 9 rarr; A 6 9 rarr; A 3
Now, substitute A 3 back into B 6 - 3A to find B:
B 6 - 3(3) 6 - 9 -3
Substitute A 3 and B -3 into equation 1 to find C:
3 (-3) C 1 rarr; C 1
Hence, the formula for the nth term is:
an 3n2 - 3n 1
Finding the Next Term
To find the next term (when n 6):
a6 3(6)2 - 3(6) 1 3(36) - 18 1 108 - 18 1 91
So, the next term in the sequence is 91.
Additional Insights
This sequence is closely related to the concept of hexagonal numbers. The sequence 1, 7, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397, 469, 547, 631, 721, 817, 919, reveals the hexagonal numbers, which are figurate numbers that represent a hexagon with a dot at each corner and all other dots surrounding the previous layer.
Related Keywords
sequence pattern quadratic formula hexagonal numbersUnderstanding this sequence not only helps in solving similar problems but also opens the door to the world of figurate numbers and their properties.