Problem 14 - Applied Discrete Structures

Question

If \(S(n)=\sum_{j=1}^{n} g(j), n \geq 1,\) then \(S\) can be described with the recurrence relation \(S(n)=S(n-1)+g(n)\). For each of the following sequences that are defined using a summation, find a closed form expression: (a) \(S(n)=\sum_{j=1}^{n} j, n \geq 1\) (b) \(Q(n)=\sum_{j=1}^{n} j^{2}, n \geq 1\) (c) \(P(n)=\sum_{j=1}^{n}\left(\frac{1}{2}\right)^{j}, n \geq 0\) (d) \(T(n)=\sum_{j=1}^{n} j^{3}, n \geq 1\)

Step-by-Step Solution

Answer

(a) \(\frac{n(n+1)}{2}\), (b) \(\frac{n(n+1)(2n+1)}{6}\), (c) \(1-\left(\frac{1}{2}\right)^n\), (d) \(\left(\frac{n(n+1)}{2}\right)^2\)."

Step 1: Analyze Sequence (a)

The sequence is defined as \( S(n) = \sum_{j=1}^{n} j \), which is the sum of the first \( n \) natural numbers. This is a well-known formula.

Step 2: Derive Formula for Sequence (a)

The closed form for \( S(n) = \sum_{j=1}^{n} j \) is \( \frac{n(n+1)}{2} \). This can be derived by considering the series as \( 1 + 2 + 3 + \ldots + n \) and noting the arithmetic progression.

Step 3: Analyze Sequence (b)

The sequence is \( Q(n) = \sum_{j=1}^{n} j^2 \), which is the sum of the squares of the first \( n \) natural numbers. This sequence has a known closed form.

Step 4: Derive Formula for Sequence (b)

The closed form for the sum of squares \( Q(n) = \sum_{j=1}^{n} j^2 \) is \( \frac{n(n+1)(2n+1)}{6} \). This can be established using mathematical induction or proven through algebraic manipulation.

Step 5: Analyze Sequence (c)

The sequence is \( P(n) = \sum_{j=1}^{n}\left(\frac{1}{2}\right)^{j} \), which is a geometric series with the common ratio \( r = \frac{1}{2} \).

Step 6: Derive Formula for Sequence (c)

The sum of the first \( n \) terms of a geometric series with ratio \( r \) is \( \frac{r(1-r^n)}{1-r} \). Substituting \( r = \frac{1}{2} \), the closed form for \( P(n) \) is \( 1 - \left(\frac{1}{2}\right)^n \).

Step 7: Analyze Sequence (d)

The sequence \( T(n) = \sum_{j=1}^{n} j^3 \) represents the sum of cubes of the first \( n \) natural numbers. The closed form for this sequence is based on a known formula.

Step 8: Derive Formula for Sequence (d)

The closed form for the sum of cubes \( T(n) = \sum_{j=1}^{n} j^3 \) is \( \left(\frac{n(n+1)}{2}\right)^2 \). This formula can be proven by considering that the sum of cubes is the square of the sum of the first \( n \) natural numbers.

Key Concepts

Summation

Summation plays a fundamental role in mathematics, acting as the cornerstone for many complex topics. When you see summation notation, \( \sum \), it represents the addition of a sequence of numbers. More formally, it describes the sum of terms as they vary, usually indicated by an index. In our case, it's taking a series of numbers from 1 to \( n \).

For example, \( \sum_{j=1}^{n} j \) means that you're adding all integers from 1 up to \( n \). Summation is often the first step in recognizing patterns or deriving formulas, as it allows us to see the relationship between terms. It is through summation that we approach the idea of closed form expressions, which simplify these repeated additions into a single, manageable formula.

Understanding the basics of summation is crucial when dealing with sequences and series. It sets the groundwork for further exploration into advanced mathematical concepts like closed form expressions and series patterns.

Closed Form Expression

A closed form expression is a crucial concept when dealing with the sum of series. It's a mathematical expression that allows you to calculate the result of a series without needing to perform the summation each time. Instead of adding each term individually, you use an equation to find the result directly.

For instance, the sequence \( S(n) = \sum_{j=1}^{n} j \), which adds all natural numbers from 1 to \( n \), can be simplified to a closed form expression \( \frac{n(n+1)}{2} \). Once plugged into this formula, you can quickly compute the sum without adding each number manually.

Closed form expressions simplify complex calculations. They save time and reduce errors, making them especially useful in solving real-world problems where efficiency is key. Whether dealing with arithmetic or geometric series, finding a closed form provides a valuable shortcut.

Arithmetic Progression

Arithmetic progression, or arithmetic sequence, is a pattern where each term after the first is the result of adding a constant to the previous term. This constant is called the "common difference." For example, in the sequence 1, 3, 5, 7, ..., the common difference is 2.

The standard formula for the sum of the first n terms of an arithmetic sequence is \( \frac{n}{2}(a + l) \), where \(a\) is the first term and \(l\) is the n-th term. This can also be rewritten using the common difference: \( \frac{n}{2}(2a + (n-1)d) \), where \(d\) is the common difference.

Arithmetic progressions are everywhere in mathematics. They help explain phenomena ranging from simple counting situations to complex financial calculations. Understanding the fundamentals of arithmetic progressions enables you to spot patterns and make predictions about sums and sequences.

Geometric Series

A geometric series is one where each term is found by multiplying the previous term by a constant, which is the "common ratio." For instance, in a series like 2, 4, 8, 16, ..., the common ratio is 2. Geometric series often arise in scenarios involving exponential growth or decay.

The formula for the sum of the first n terms of a geometric series is \( \frac{a(1-r^n)}{1-r} \), where \(a\) is the first term and \(r\) is the common ratio. This formula allows you to easily compute the total sum without having to multiply out each term.

Understanding geometric series is essential in many fields, such as finance for calculating compound interest or in computer science when optimizing recursive functions. The concept is key to mastering many forms of mathematical and practical problem solving.