Question

[T] Consider a series combining geometric growth and arithmetic decrease. Let \(a_{1}=1 .\) Fix \(a>1\) and \(0

Short answer

The formula is \( a_{n+1} = a^n - \frac{b}{a-1} (a^n - 1) \); for convergence, \( b \geq a-1 \).

Step-by-step solution

Define Recursive Relation

The given recursive relation is \( a_{n+1} = a \cdot a_n - b \). We will use this to derive a formula for \( a_n \) by expressing it in terms of the initial condition and expressing it generally.

Expand the Terms

Write out the first few terms to observe the pattern. Starting with \( a_1 = 1 \), we have: \( a_2 = a \cdot a_1 - b = a - b \) \( a_3 = a \cdot a_2 - b = a(a-b) - b = a^2 - ab - b \) Continue this process to identify a pattern for \( a_{n+1} \).

Identify the Pattern

Notice a recurrence in the structure of the equation: every term includes powers of \( a \), subtracted with multiples of \( b \). The terms expand into a series where each step involves multiplying by \( a \) and adjusting with \( -b \cdot (n-1) \).

Generalize the Formula

Assume a general pattern \( a_{n+1} = a^n - b \cdot (a^{n-1} + a^{n-2} + \ldots + 1) \). This comes from expanding each term recursively as seen in earlier steps. This summation can be expressed as a geometric series.

Simplify Using Geometric Series Formula

The expression \( a^{n-1} + a^{n-2} + \ldots + 1 \) is a geometric series with ratio \( a \) and \( n \) terms. The sum of this series is \( \frac{a^n - 1}{a - 1} \). Thus, \( a_{n+1} = a^n - b \cdot \frac{a^n - 1}{a - 1} \).

Determine Convergence Condition

For \( a_n \) to converge, \( a_{n+1} \) must lead to a stable limit, meaning the series must not infinitely grow or oscillate. The sequence converges when \( |a| < 1 \) or, for the combination of growth and subtraction to stabilize, when \( b \geq a - 1 \).

Final Formula and Condition

Thus, the formula for \( a_{n+1} \) is \( a_{n+1} = a^n - \frac{b}{a-1} (a^n - 1) \) and the condition for convergence is when \( b \geq a - 1 \). This ensures that the net effect of the geometric growth and arithmetic decrease results in convergence.

Key concepts

Recursive Relations

In mathematics, a **recursive relation** is a formula that describes the relationship between terms in a sequence. At its core, it provides a way to define each term based on its preceding terms. This iterative process continues by relating each new term to the previous ones, establishing a pattern. This type of relation is particularly useful for sequences where direct formula derivation is challenging.

For the exercise given, the recursive relation is:

• \( a_{n+1} = a \cdot a_n - b \)

This formula indicates that to find the next term \( a_{n+1} \), you multiply the current term \( a_n \) by a constant factor \( a \) and then subtract a constant value \( b \).
Understanding recursive relations is pivotal in identifying underlying patterns in sequences, which eventually leads to deriving a general formula. This kind of formula often uncovers the progression or decline in series such as those involving growth and decay. By systematically applying the recursive relation, you can expand and transform it into a pattern, gradually leading to simplification and formula discovery as seen in this exercise.

Convergence Criteria

The concept of **convergence criteria** revolves around conditions under which a series or sequence approaches a stable limit. In other words, it determines whether or not the sequence will settle at a specific value as the number of terms grows indefinitely.

For the sequence in the original problem, convergence depends on how the values a and b are chosen. Specifically, the sequence will converge when:

• The growth factor \( a \) is kept in balance with the constant subtraction term \( b \).
• As derived: \( b \geq a - 1 \) ensures the series does not expand uncontrollably.

The significance is that convergence ensures that even as more terms are added, the sequence stabilizes rather than oscillating, increasing, or decreasing indefinitely. A deep understanding of convergence criteria is integral for predicting long-term behavior of mathematical models and sequences.

When dealing with convergence in mathematical problems, it's useful to visualize it as your series 'aiming' to reach steady state, making mathematical analysis and problem-solving relatable to real-world phenomena such as calculating investment growth or modeling population dynamics.

Arithmetic Decrease

An **arithmetic decrease** involves deducting a constant value from a quantity repeatedly. This concept is approachable when seen as a fixed decrement applied across intervals. In sequence and series problems, arithmetic decrease offers a balancing force, countering growth or expansion.

In the sequence given, arithmetic decrease is represented by the subtraction of \( b \) throughout the equation:

• \( a_{n+1} = a \cdot a_n - b \)

This arithmetic component ensures that despite the potential exponential growth derived from multiplication by \( a \), the subtraction of \( b \) tempers the increase.

Examining arithmetic decrease, in conjunction with geometric terms, suggests a careful balance orchestrated to control expansion. In practical scenarios, understanding the impact of an arithmetic decrease helps with predictions in gradual declines, investment depreciation, or interest compensation. It's a foundational principle that, when combined with other mathematical operations, leads to a rich analysis of sequences in both theoretical and applied mathematics.