Problem 21 - Discrete Mathematics with Applications

Question

Using Exercise 19 and the identity \((i+1)^{2}-i^{2}=2 i+1,\) find a formula for $$\sum_{i=1}^{n} i$$

Step-by-Step Solution

Answer

Short Answer: The formula for the sum of integers from 1 to n is \(\sum_{i=1}^{n} i = \frac{(n+1)^{2}-n-1}{2}\).

Step 1: 1. Sum the sequence of i

Let S be the sum of integers from 1 to n, then \[S = 1 + 2 + 3 + \cdots + (n-1) + n\]

Step 2: 2. Apply the given identity

The identity given is \((i+1)^{2} - i^{2} = 2i + 1\). We will apply this identity for each i in the range 1 to n.

Step 3: 3. Apply the identity to each term of the sum

Now let's apply the identity to each term in the sum: \((2^{2} - 1^{2}) + (3^{2} - 2^{2}) + (4^{2} - 3^{2}) + \cdots + ((n+1)^{2} - n^{2})\) We notice that each term now has the difference between consecutive squares and matches the given identity.

Step 4: 4. Simplify the obtained expression

The expression from step 3 can be rearranged and simplified: \((2^{2} - 1^{2}) + (3^{2} - 2^{2}) + \cdots + ((n+1)^{2} - n^{2}) = (n+1)^{2} - 1^{2}\]

Step 5: 5. Relate the simplified expression to the sum formula

By step 2, the sum S also equals the expression from step 4: \[S = (n+1)^{2} - 1^{2}\] However, by applying the identity given, we also have that: \[(n+1)^{2} - 1^{2} = 2(1) + 1 + 2(2) + 1 + \ldots + 2(n) + 1\] Therefore, S can also be represented as: \[S = \sum_{i=1}^{n} (2i + 1)\]

Step 6: 6. Determine the sum formula

Now, we can rewrite the expression from step 5: \[S = \sum_{i=1}^{n} (2i + 1) = 2\left(\sum_{i=1}^{n} i \right) + n\] Solve for \(\sum_{i=1}^{n} i\): \[\sum_{i=1}^{n} i = \frac{S-n}{2} = \frac{((n+1)^{2} - 1^2) - n}{2}\] Thus, the formula for the sum of integers from 1 to n is \[\boxed{\sum_{i=1}^{n} i = \frac{(n+1)^{2}-n-1}{2}}\]

Key Concepts

Understanding Mathematical Identity

A mathematical identity is an equation that holds true for all possible values of its variables. In the context of summing integers, we often use identities to simplify expressions or to reveal underlying patterns. For instance, consider the identity \((i+1)^2 - i^2 = 2i + 1\). This identity expresses the difference between consecutive squares.

• This specific identity allows us to break down complex sums into simpler terms.
• It's a universal truth, meaning it consistently holds regardless of the value of \(i\).

By applying such math identities, we can transform and solve problems more efficiently. Here, we used it to derive the sum of integers formula, which typically involves tackling a series in a step-by-step manner, eventually boiling it down to a simple expression.

Exploring Finite Series

Finite series is the sum of the terms of a sequence up to a fixed limit \(n\). When you sum integers from 1 to \(n\), you're dealing with a classic example of a finite series. This series is often symbolically represented as \(\sum_{i=1}^{n} i\) and can be understood through the sequential addition of numbers.

• Each term is an individual integer increasing incrementally by 1.
• The total or sum is only over these finite specified terms, and not beyond.

To find a formula for this sum, mathematical identities can be key. As demonstrated, applying \((2i + 1)\) through the given identity enables us to express the series in terms of a known formula. This connection highlights how identities simplify the computations involved with finite series.

Delving into Discrete Mathematics

Discrete mathematics is the branch of math dealing with countable, distinct elements. It's all about distinct and separate values or objects. This exercise is rooted in discrete mathematics because it deals with the sum of integers, which are discrete elements.

• Unlike continuous mathematics, discrete math focuses on such separated and non-continuous data.
• Summing integers is a classic problem within this field, often tackled using sequences and series.

The use of discrete variables (like integers) calls for specific strategies to navigate these finite structures. By employing formulas and identities, discrete math provides robust tools to address problems, like finding the sum of integers, efficiently. These approaches simplify the process of computing values that otherwise might take multiple individual calculations.