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An increasing function is a type of function where the value of the function increases as the input value increases. Mathematically, a function \(f(x)\) is increasing if for any two numbers \(x_1\) and \(x_2\) such that \(x_1 < x_2\), the inequality \(f(x_1) \leq f(x_2)\) holds. This characteristic is important because it means the function does not decrease; it can only stay the same or grow.
In our context, the function \(f\) adheres to the recurrence relation \(f(n) = a f(n / b) + c n^d\). The fact that \(f\) is increasing implies that as \(n\) grows, the value of \(f(n)\) also grows, albeit in a structured way dictated by the recurrence relation.
Understanding increasing functions helps us grasp how the parameters \(a\), \(b\), \(c\), and \(d\) influence the growth of \(f(n)\). By knowing that \(a\geq 1\), we ensure that each step in the recurrence relation keeps \(f(n)\) from decreasing. The term \(c n^d\) adds a positive component, further promoting the increase.
When analyzing recurrence relations, we often start by verifying if the functions involved are increasing, as it simplifies the mathematical manipulation and ensures logical consistency.

The power of a number refers to the operation of multiplying a number by itself a given number of times. For example, \(n^d\) means \(n\) is multiplied by itself \(d\) times. In our problem, the power function \(n^d\) appears directly in the recurrence relation and in the final solution.
Here's a recap of key properties:

• Multiplicative Identity: Any number raised to the power of 1 is the number itself, i.e., \(n^1 = n\).
• Zero Property: Any number (except zero) raised to the power of 0 is 1, i.e., \(n^0 = 1\).
• Product of Powers: When multiplying two powers with the same base, add the exponents, \(n^a \cdot n^b = n^{a+b}\).

In our recurrence relation, when we substitute \(n = b^k\), the term \(c n^d\) becomes \(c(b^k)^d = c b^{kd}\).
Recognizing that \(n\) is a power of \(b\) (e.g., \(n = b^k\)) allows us to simplify the recurrence relation step-by-step. It acknowledges the geometric structure of the powers and employs it in solving the recurrence relation efficiently.

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. For example, the series \(1, r, r^2, r^3, \dots\) is geometric with common ratio \(r\).
In our solution, we encounter a finite geometric series when expressing \(f(n)\). To display \the step 4 result of our recurrence relation, we use the series: \(1 + b^{-d} + b^{-2d} + \dots \dots + b^{-(k-1)d}\). This series can be summed up using: \If the first term of the series is \(a\) and the common ratio is \(r\), the sum of the first \(n\) terms is given by:

\[S_n = a \frac{1-r^n}{1-r}\]
Applying this to our series, \(a = 1\), \(r = \b^{-d}\) and the number of terms is \(k\), thus:
\[1 + b^{-d} + b^{-2d} + \dots + b^{-(k-1)d} = \frac{1-b^{-kd}}{1-b^{-d}}\]
Recognizing that \(n\) is \(b^k\) simplifies further as:\(b^{-kd} = (b^k)^{-d} = n^{-d}\). Therefore, substituting this geometric series back into our function explains part of \(f(n)\). We simplify it to get to our final result.\ This understanding shows the vital role geometric series play in sum-related problems, especially when dealing with recurrence relations.