Part 1/6:

Exploring a Recurrence Relation: The Sequence ( a_n )

In the realm of mathematics, sequences defined by recurrence relations are central to understanding various concepts. This article delves into the intriguing sequence ( a_n ) governed by a specific recurrence relation that will be rigorously analyzed using mathematical induction and limit properties.

The Recurrence Relation

The recurrence relation defining the sequence ( a_n ) is as follows:

( a_1 = 2 )
( a_{n+1} = \frac{12 a_n + 6}{2} ) for ( n ) being a positive integer (1, 2, 3, ...).

To understand how this sequence behaves, we begin by computing the first few terms.

Calculating Initial Terms

Let's calculate the first several terms to observe any patterns:

( a_1 = 2 )
For ( a_2 ):

[