The 1, 5, 13, 25 Sequence: Unraveling the Formula Behind the Pattern
The 1, 5, 13, 25 sequence is a fascinating pattern that has intrigued mathematicians and puzzle enthusiasts alike. At first glance, the numbers may seem random, but upon closer inspection, a hidden formula emerges. In this article, we'll delve into the world of number sequences and uncover the formula behind this intriguing pattern.
The Pattern
The sequence begins with the numbers 1, 5, 13, and 25. At first, it may appear to be a random assortment of numbers, but as we examine the sequence more closely, a pattern begins to emerge.
1, 5, 13, 25, ...
The Formula
After observing the sequence, we can identify a recurring pattern. Each term is obtained by adding a fixed constant to the previous term. This constant is none other than 4, 8, 12, and so on. Do you see the pattern?
1 + 4 = 5 5 + 8 = 13 13 + 12 = 25
The formula behind the sequence can be expressed as:
an = an-1 + (4n - 4)
where an is the nth term of the sequence, and n is a positive integer (1, 2, 3, ...).
Derivation of the Formula
Let's derive the formula step by step:
- The sequence starts with 1, so a1 = 1.
- The difference between the first two terms is 4 (5 - 1 = 4).
- The difference between the second and third terms is 8 (13 - 5 = 8).
- The difference between the third and fourth terms is 12 (25 - 13 = 12).
By observing the pattern of differences (4, 8, 12, ...), we can conclude that the nth difference is 4n - 4.
Therefore, the formula for the sequence is:
an = an-1 + (4n - 4)
Conclusion
The 1, 5, 13, 25 sequence, at first glance, may seem like a random collection of numbers. However, upon closer inspection, a hidden formula emerges. The sequence is generated by adding a fixed constant to the previous term, with the constant increasing by 4 each time. The formula for the sequence is an = an-1 + (4n - 4), where an is the nth term of the sequence, and n is a positive integer.
With this newfound understanding, we can generate the next terms of the sequence with ease:
1, 5, 13, 25, 41, 61, ...
The possibilities are endless, and the beauty of mathematics lies in its ability to uncover hidden patterns and structures in seemingly random sequences.
