Deciphering the Sequence 1, 5, 17, 53, 161, 485: Understanding the Pattern and Rule

Sequences are a fascinating aspect of mathematics, often hidden within patterns and rules that require careful observation. In this article, we will explore one such intriguing sequence: 1, 5, 17, 53, 161, 485. We will delve into its underlying pattern, derive a recursive formula, and understand why the next number in the sequence is 485.

The Pattern Unveiled

The sequence provided is 1, 5, 17, 53, 161, 485. To understand the rule governing this sequence, let's break it down step by step.

Step-by-Step Analysis

Observe the transformation from one number to the next in the sequence:

1 (start) x 3 2 5 5 x 3 2 17 17 x 3 2 53 53 x 3 2 161 161 x 3 2 485

Each term follows a specific rule: multiply the previous term by 3 and then add 2.

General Recursive Formula

We can express this recursively as:

Recursive Formula:

a_{n 1} 3 cdot a_n 2

With the initial term:

Initial Term:

a_1 1

General Term Expression

To find a general term expression for the sequence, let's solve the recurrence relation:

a_1 1

a_{n 1} 3 cdot a_n 2

Let's assume the general term a_n can be written as:

a_n A cdot 3^n B

By substituting and solving, we get:

a_1 A cdot 3^1 B 1 a_{n 1} A cdot 3^{n 1} B 3A cdot 3^n B

We find that:

A 2/3

B -1

Thus, the general term is:

a_n (2/3) cdot 3^n - 1

Verification

Let's verify the formula with the given terms:

a_1 (2/3) cdot 3^1 - 1 2 - 1 1 a_2 (2/3) cdot 3^2 - 1 6 - 1 5 a_3 (2/3) cdot 3^3 - 1 18 - 1 17 a_4 (2/3) cdot 3^4 - 1 54 - 1 53 a_5 (2/3) cdot 3^5 - 1 162 - 1 161 a_6 (2/3) cdot 3^6 - 1 486 - 1 485

The formula holds true for all the given terms.

Additional Sequences for Comparison

Let's consider another sequence that follows a similar pattern but with different ratios and additions:

Sequence 2: 1, 2, 5, 15, 52.5, 210

In this sequence, the ratio between successive terms is increasing by 0.5 each time:

2/1 2 5/2 2.5 15/5 3 52.5/15 3.5 210/52.5 4

The next ratio is 4.5, so the next term is:

210 times 4.5 945

Conclusion

The sequence 1, 5, 17, 53, 161, 485 follows a specific pattern, where each term is obtained by multiplying the previous term by 3 and adding 2. This recursive formula and the derived general term expression provide a clear understanding of the sequence and allow for accurate prediction of subsequent terms.