Problem 1 - Applied Discrete Structures

Question

Calculate the following series: (a) \(\sum_{i=1}^{3}(2+3 i)\) (c) \(\sum_{j=0}^{n} 2^{j}\) for \(n=1,2,3,4\) (b) \(\sum_{i=-2}^{1} i^{2}\) (d) \(\sum_{k=1}^{n}(2 k-1)\) for \(n=1,2,3,4\)

Step-by-Step Solution

Answer

(a) 24, (c) 2, 4, 8, 16, (b) 6, (d) 1, 4, 9, 16

Step 1: Calculate the terms of series (a)

The expression for series (a) is \( \sum_{i=1}^{3}(2+3i) \). Begin by calculating the terms: - For \( i=1 \), the term is \( 2 + 3(1) = 5 \).- For \( i=2 \), the term is \( 2 + 3(2) = 8 \).- For \( i=3 \), the term is \( 2 + 3(3) = 11 \).

Step 2: Sum the terms of series (a)

Now, add the calculated terms from step 1: \( 5 + 8 + 11 = 24 \). So, the value of series (a) is 24.

Step 3: Recognize series (c) as a geometric series

The expression for series (c) is \( \sum_{j=0}^{n} 2^{j} \). This is a geometric series with a common ratio of 2, starting from \( j=0 \).

Step 4: Use formula to calculate series (c) for each n

The sum of a geometric series \( \sum_{j=0}^{n} a r^{j} \) is calculated by the formula \( S = a \frac{r^{n+1}-1}{r-1} \). For this series:- \( a = 1, r = 2 \).- For \( n=1 \), \( S = 2 \).- For \( n=2 \), \( S = 4 \).- For \( n=3 \), \( S = 8 \).- For \( n=4 \), \( S = 16 \).

Step 5: Determine terms of series (b)

The expression for series (b) is \( \sum_{i=-2}^{1} i^{2} \). Calculate each term:- For \( i=-2 \), the term is \( (-2)^{2} = 4 \).- For \( i=-1 \), the term is \( (-1)^{2} = 1 \).- For \( i=0 \), the term is \( 0^{2} = 0 \).- For \( i=1 \), the term is \( 1^{2} = 1 \).

Step 6: Sum the terms of series (b)

Add the calculated terms from step 5: \( 4 + 1 + 0 + 1 = 6 \). So, series (b) equals 6.

Step 7: Recognize series (d) as a series of odd numbers

The expression for series (d) is \( \sum_{k=1}^{n}(2k-1) \). This represents the sum of the first \( n \) odd numbers, which can be expressed as \( n^{2} \).

Step 8: Calculate series (d) using the formula for each n

Use the formula \( n^{2} \) to find the sum:- For \( n=1 \), \( 1^{2} = 1 \).- For \( n=2 \), \( 2^{2} = 4 \).- For \( n=3 \), \( 3^{2} = 9 \).- For \( n=4 \), \( 4^{2} = 16 \).

Key Concepts

Geometric Series

A geometric series is a fascinating concept where each term after the first is found by multiplying the previous term by a constant, known as the common ratio. In mathematical terms, it can be expressed as \( a, ar, ar^2, ar^3, \ldots \), where \( a \) is the first term and \( r \) is the common ratio.
Understanding geometric series is crucial because they frequently appear in both pure and applied mathematics. They can be finite or infinite.

To calculate the sum of a finite geometric series, we use the formula:

• \( S_n = a \frac{r^{n+1} - 1}{r-1} \)

For series (c) from the exercise, \( a = 1 \), \( r = 2 \), and \( n \) varies from 1 to 4. For example, when \( n = 4 \), applying the formula gives \( S_4 = 2^{4} = 16 \). It demonstrates a simple way to find out cumulative values in problems involving multiplicative growth.
This makes understanding geometric series highly valuable, especially in fields involving exponential growth such as financial forecasting or population studies.

Sum of Odd Numbers

The sum of odd numbers results in a surprisingly simple and beautiful pattern. Specifically, the sum of the first \( n \) odd numbers is given by the formula \( n^2 \). Each odd number can be represented in the form \( (2k-1) \), where \( k \) is any integer starting from 1.
This characteristic is crucial for quickly computing totals without adding each term individually. In exercise (d), you're asked to calculate the sum of odd numbers up to a certain \( n \). When \( n = 4 \), for example, the sum of the first four odd numbers (1, 3, 5, 7) is calculated as \( 4^2 = 16 \).
Some engaging insights about why this works involve visualization and rearranging pebbles or stones in a square, matching each successive odd number to fill the next layer of the square. This relationship often surprises learners with its simplicity and symmetry, making it a delightful discovery to appreciate the elegance of mathematics.

Series Calculation Steps

Calculating a series involves breaking down expressions into individual terms and combining these terms to reach a total sum. This process requires attentive handling of mathematical expressions.
Let's take series (b) from the exercise as an example. It's given by the expression \( \sum_{i=-2}^{1} i^2 \). The task is to find the square of each integer from -2 to 1 and then sum these.
The steps are as follows:

• Calculate each squared term:
  • For \( i = -2 \), \( (-2)^2 = 4 \)
  • For \( i = -1 \), \( (-1)^2 = 1 \)
  • For \( i = 0 \), \( 0^2 = 0 \)
  • For \( i = 1 \), \( 1^2 = 1 \)
• Sum the calculated terms: \( 4 + 1 + 0 + 1 = 6 \)

Each calculation step adds to understanding how series work. It's about moving meticulously from one calculated piece to another until the whole is assembled. These practices are common in many branches of mathematics and are key for analyzing sequences and measuring changes over time.